Editing Appendix/Ramblings/BiPolytrope51AnalyticStability (section)

=====Attempt 3=====
======Straightforward Trial======
Let's adopt a trial eigenfunction of the form
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_\mathrm{trial}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b_0}{\Lambda^2} \biggl[
1 - \Lambda \cot(\Lambda - E)
\biggr] - a_0 \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Lambda</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Lambda_0 + g_\mathrm{F} \eta \, .</math>
  </td>
</tr>
</table>

<table border="1" width="60%" align="center" cellpadding="8"><tr><td align="left">
<font color="red">'''NOTE:'''</font>&nbsp; We can retrieve the empirical expression for <math>~x_\mathrm{trial}</math> obtained above in '''Attempt 2''' if we eventually set,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Lambda_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta_i + g_\mathrm{F}(\eta_i - 2\eta_s)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~g_\mathrm{F}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\pi}{8(\eta_s - \eta_i)} \, ,</math> &nbsp; &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~E</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta_i - \frac{5\pi}{4} + \tan^{-1} f_\alpha \, .</math>
  </td>
</tr>
</table>
The last of these expressions arises because,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cot(\Lambda - E)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan \biggl[ \frac{\pi}{2} - (\Lambda - E)\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan\biggl[ \frac{\pi}{2} - \Lambda  + \eta_i - \frac{5\pi}{4} + \tan^{-1} f_\alpha \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan\biggl[ (\eta_i - \Lambda - \tfrac{3\pi}{4}) + \tan^{-1}f_\alpha \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\tan(\eta_i - \Lambda - \tfrac{3\pi}{4}) + f_\alpha }{1 - f_\alpha \tan(\eta_i - \Lambda - \tfrac{3\pi}{4} )} \, .</math>
  </td>
</tr>
</table>

</td></tr></table>


Because the LAWE requires derivatives of <math>~x_\mathrm{trial}</math> with respect to <math>~\eta</math>, we will often need to recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d\Lambda}{d\eta} \cdot \frac{d}{d\Lambda} = g_\mathrm{F} \cdot \frac{d}{d\Lambda} \, .</math>
  </td>
</tr>
</table>
Hence, in particular,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{d\eta}\biggl[\cot(\Lambda - E) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- g_\mathrm{F} [1 + \cot^2(\Lambda - E) ] \, .</math>
  </td>
</tr>
</table>
The first derivative gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{d\eta}\biggl[ x_\mathrm{trial} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\eta} \biggl\{
\frac{b_0}{\Lambda^2} \biggl[
1 - \Lambda \cot(\Lambda - E)
\biggr] - a_0 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{2b_0 g_\mathrm{F}}{\Lambda^3} \biggl[1 - \Lambda \cot(\Lambda - E)\biggr] 
+ \frac{b_0}{\Lambda^2} \biggl[- g_\mathrm{F} \cot(\Lambda - E)\biggr] 
+ \frac{b_0}{\Lambda^2} \biggl[g_\mathrm{F} \Lambda [1 + \cot^2(\Lambda - E) ] \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ g_\mathrm{F} b_0 \biggl\{
-~\frac{2}{\Lambda^3} \biggl[1 - \Lambda \cot(\Lambda - E)\biggr] 
- \frac{1}{\Lambda^2} \biggl[\cot(\Lambda - E)\biggr] 
+ \frac{1}{\Lambda} \biggl[1 + \cot^2(\Lambda - E) \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ g_\mathrm{F} b_0 \biggl\{
\frac{1}{\Lambda}
-\frac{2}{\Lambda^3}  
+\frac{1}{\Lambda^2} \biggl[\cot(\Lambda - E)\biggr] 
+ \frac{1}{\Lambda} \biggl[\cot^2(\Lambda - E) \biggr] 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

The second derivative gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{g_\mathrm{F}b_0} \cdot \frac{d^2}{d\eta^2}\biggl[ x_\mathrm{trial} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d}{d\eta}\biggl\{
\frac{1}{\Lambda}
-\frac{2}{\Lambda^3}  
+\frac{1}{\Lambda^2} \biggl[\cot(\Lambda - E)\biggr] 
+ \frac{1}{\Lambda} \biggl[\cot^2(\Lambda - E) \biggr] 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{g_\mathrm{F}}{\Lambda^2}
+\frac{6 g_\mathrm{F}}{\Lambda^4}  
-\frac{2g_\mathrm{F}}{\Lambda^3} \biggl[\cot(\Lambda - E)\biggr] 
- \frac{g_\mathrm{F}}{\Lambda^2} \biggl[1 + \cot^2(\Lambda - E)\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{g_\mathrm{F}}{\Lambda^2} \biggl[\cot^2(\Lambda - E) \biggr] 
- \frac{2g_\mathrm{F}}{\Lambda} \biggl[\cot(\Lambda - E) \biggr] \biggl[1 + \cot^2(\Lambda - E) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{1}{g^2_\mathrm{F}b_0} \cdot \frac{d^2}{d\eta^2}\biggl[ x_\mathrm{trial} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{1}{\Lambda^2}
+\frac{6 }{\Lambda^4}  
-\frac{2}{\Lambda^3} \biggl[\cot(\Lambda - E)\biggr] 
- \frac{1}{\Lambda^2} 
- \frac{1}{\Lambda^2} \biggl[\cot^2(\Lambda - E)\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{1}{\Lambda^2} \biggl[\cot^2(\Lambda - E) \biggr] 
- \frac{2}{\Lambda} \biggl[\cot(\Lambda - E)  + \cot^3(\Lambda - E) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{1}{\Lambda^2}
- \frac{1}{\Lambda^2} 
+\frac{6 }{\Lambda^4}  
-\frac{2}{\Lambda^3} \biggl[\cot(\Lambda - E)\biggr] 
- \frac{2}{\Lambda} \biggl[\cot(\Lambda - E)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{1}{\Lambda^2} \biggl[\cot^2(\Lambda - E) \biggr] 
- \frac{1}{\Lambda^2} \biggl[\cot^2(\Lambda - E)\biggr] 
- \frac{2}{\Lambda} \biggl[ \cot^3(\Lambda - E) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{2}{\Lambda^2}
+\frac{6 }{\Lambda^4}  
- \biggl[ \frac{2}{\Lambda^3} + \frac{2}{\Lambda}\biggr]\biggl[\cot(\Lambda - E)\biggr] 
- \frac{2}{\Lambda^2} \biggl[\cot^2(\Lambda - E) \biggr] 
- \frac{2}{\Lambda} \biggl[ \cot^3(\Lambda - E) \biggr] \, .
</math>
  </td>
</tr>
</table>
So, appreciating that, <math>~\eta = (\Lambda - \Lambda_0 )/g_\mathrm{F}</math>, and dividing the relevant LAWE through by <math>~b_0 g_\mathrm{F}^2</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{b_0 g_\mathrm{F}^2} \cdot 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{b_0 g_\mathrm{F}^2} \cdot \frac{d^2x_\mathrm{trial} }{d\eta^2} 
+ \frac{1}{b_0 g_\mathrm{F}^2} \biggl[ 2 - Q \biggr] \frac{2g_\mathrm{F}}{(\Lambda - \Lambda_0 )}\cdot \frac{dx_\mathrm{trial}}{d\eta} 
- \frac{2Q}{b_0 g_\mathrm{F}^2} \cdot  \frac{g^2_\mathrm{F} x_\mathrm{trial}}{(\Lambda - \Lambda_0 )^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(\Lambda - \Lambda_0 )^2} \biggl\{
\frac{(\Lambda - \Lambda_0 )^2}{b_0 g_\mathrm{F}^2} \cdot \frac{d^2x}{d\eta^2} 
+ \frac{ (\Lambda - \Lambda_0 )}{b_0 g_\mathrm{F}} \biggl[ 2 - Q \biggr] \frac{dx}{d\eta} 
- \frac{2Q}{b_0 } \cdot  x
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{ (\Lambda - \Lambda_0 )^2}{b_0 g_\mathrm{F}^2} \cdot 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\Lambda - \Lambda_0 )^2 \biggl\{ \frac{1}{b_0 g_\mathrm{F}^2} \cdot \frac{d^2x}{d\eta^2} \biggr\}
+ (\Lambda - \Lambda_0 ) \biggl[ 4 - 2Q \biggr] \biggl\{ \frac{1}{b_0 g_\mathrm{F}} \frac{dx}{d\eta} \biggr\}
- 2Q \biggl\{ \frac{x}{b_0} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\Lambda - \Lambda_0 )^2 \biggl\{ 
-\frac{2}{\Lambda^2}
+\frac{6 }{\Lambda^4}  
- \biggl[ \frac{2}{\Lambda^3} + \frac{2}{\Lambda}\biggr]\biggl[\cot(\Lambda - E)\biggr] 
- \frac{2}{\Lambda^2} \biggl[\cot^2(\Lambda - E) \biggr] 
- \frac{2}{\Lambda} \biggl[ \cot^3(\Lambda - E) \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\Lambda - \Lambda_0 ) \biggl[ 4 - 2Q \biggr] \biggl\{ \frac{1}{\Lambda}
-\frac{2}{\Lambda^3}  
+\frac{1}{\Lambda^2} \biggl[\cot(\Lambda - E)\biggr] 
+ \frac{1}{\Lambda} \biggl[\cot^2(\Lambda - E) \biggr] 
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 2Q \biggl\{ \frac{1}{\Lambda^2} \biggl[1 - \Lambda \cot(\Lambda - E) \biggr] - \frac{ a_0 }{b_0}
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{ \Lambda^4(\Lambda - \Lambda_0 )^2}{b_0 g_\mathrm{F}^2} \cdot 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\Lambda - \Lambda_0 )^2 \biggl\{ 
(6-2\Lambda^2)
- (2\Lambda + 2\Lambda^3 ) \cot(\Lambda - E) 
- 2\Lambda^2 \cot^2(\Lambda - E)  
- 2\Lambda^3 \cot^3(\Lambda - E) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\Lambda - \Lambda_0 ) (4 - 2Q ) \biggl\{ \Lambda^3
-2\Lambda  
+\Lambda^2 \cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 2Q \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{ \Lambda^4(\Lambda - \Lambda_0 )^2}{2b_0 g_\mathrm{F}^2} \cdot 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\Lambda - \Lambda_0 ) \biggl\{ 
(3-\Lambda^2)(\Lambda - \Lambda_0 )
- (\Lambda + \Lambda^3 )(\Lambda - \Lambda_0 ) \cot(\Lambda - E) 
- \Lambda^2 (\Lambda - \Lambda_0 )\cot^2(\Lambda - E)  
- \Lambda^3 (\Lambda - \Lambda_0 )\cot^3(\Lambda - E) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\Lambda - \Lambda_0 ) \biggl\{ 2\Lambda^3
-4\Lambda  
+2\Lambda^2 \cot(\Lambda - E) 
+ 2\Lambda^3 \cot^2(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-Q \biggl\{ \Lambda^3(\Lambda - \Lambda_0 )
-2\Lambda  (\Lambda - \Lambda_0 )
+\Lambda^2 (\Lambda - \Lambda_0 )\cot(\Lambda - E) 
+ \Lambda^3 (\Lambda - \Lambda_0 )\cot^2(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- Q \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr\} \, .
</math>
  </td>
</tr>
</table>

----

The right-hand-side of this expression should simplify considerably if we let <math>~E \rightarrow B</math>, if we set <math>~(g_\mathrm{F}, \Lambda_0) = (1, 0) ~\Rightarrow~ \Lambda = \eta</math>, and if <math>~(a_0, b_0) = (0, 3)</math>.  Let's see.

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Lambda \biggl\{ 
(3-\Lambda^2)\Lambda 
- (\Lambda + \Lambda^3 )\Lambda  \cot(\Lambda - E) 
- \Lambda^3 \cot^2(\Lambda - E)  
- \Lambda^4 \cot^3(\Lambda - E) 
+2\Lambda^3
-4\Lambda  
+2\Lambda^2 \cot(\Lambda - E) 
+ 2\Lambda^3 \cot^2(\Lambda - E)  
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-Q \biggl\{ \Lambda^4 
-2\Lambda^2  
+\Lambda^3 \cot(\Lambda - E) 
+ \Lambda^4 \cot^2(\Lambda - E)  
+\Lambda^2 - \Lambda^3 \cot(\Lambda - E)   
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Lambda \biggl\{ 
(-\Lambda +\Lambda^3) 
+ (\Lambda^2 - \Lambda^4) \cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  
- \Lambda^4 \cot^3(\Lambda - E) 
-\Lambda^3 
+ \Lambda  
- \Lambda^3 \cot^2(\Lambda - E)  
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\Lambda\cot(\lambda-E) \biggl\{ \Lambda^4 
-2\Lambda^2  
+\Lambda^3 \cot(\Lambda - E) 
+ \Lambda^4 \cot^2(\Lambda - E)  
+\Lambda^2 - \Lambda^3 \cot(\Lambda - E)   
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Lambda \biggl\{ 
(\Lambda^2 - \Lambda^4) \cot(\Lambda - E) 
- \Lambda^4 \cot^3(\Lambda - E) 
\biggr\}
+\Lambda\cot(\lambda-E) \biggl\{ \Lambda^4 
-\Lambda^2  
+ \Lambda^4 \cot^2(\Lambda - E)  
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, .
</math>
  </td>
</tr>
</table>

EXCELLENT !!

----

Let's return to the more general expression and see if it can be simplified.

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{ \Lambda^4(\Lambda - \Lambda_0 )^2}{2b_0 g_\mathrm{F}^2} \cdot 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Lambda (\Lambda - \Lambda_0 ) \biggl\{ 
(3-\Lambda^2) 
- (\Lambda + \Lambda^3 )  \cot(\Lambda - E) 
- \Lambda^2 \cot^2(\Lambda - E)  
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \Lambda_0 (\Lambda - \Lambda_0 ) \biggl\{ 
(3-\Lambda^2)
- (\Lambda + \Lambda^3 ) \cot(\Lambda - E) 
- \Lambda^2 \cot^2(\Lambda - E)  
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \Lambda (\Lambda - \Lambda_0 ) \biggl\{ 2\Lambda^2
-4 
+2\Lambda \cot(\Lambda - E) 
+ 2\Lambda^2 \cot^2(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-Q(\Lambda - \Lambda_0 ) \biggl\{ \Lambda^3
-2\Lambda 
+\Lambda^2\cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  
 \biggr\}
- Q \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Lambda (\Lambda - \Lambda_0 ) \biggl\{ 
(\Lambda^2-1) 
+ (\Lambda - \Lambda^3 )  \cot(\Lambda - E) 
+ \Lambda^2 \cot^2(\Lambda - E)  
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr\}
- \Lambda (\Lambda - \Lambda_0 ) \biggl\{ \Lambda^2
-2 
+\Lambda \cot(\Lambda - E) 
+ \Lambda^2 \cot^2(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \Lambda_0 (\Lambda - \Lambda_0 ) \biggl\{ 
(3-\Lambda^2)
- (\Lambda + \Lambda^3 ) \cot(\Lambda - E) 
- \Lambda^2 \cot^2(\Lambda - E)  
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr\}
-  \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \eta\cot(\eta-B)  (\Lambda - \Lambda_0 ) \biggl\{ \Lambda^3
-2\Lambda 
+\Lambda^2\cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  
 \biggr\}
+ \eta\cot(\eta-B)  \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Lambda (\Lambda - \Lambda_0 ) \biggl\{ 
1 
- \Lambda^3  \cot(\Lambda - E) 
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \Lambda_0 (\Lambda - \Lambda_0 ) \biggl\{ 
(3-\Lambda^2)
- (\Lambda + \Lambda^3 ) \cot(\Lambda - E) 
- \Lambda^2 \cot^2(\Lambda - E)  
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr\}
-  \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \eta\cot(\eta-B)  (\Lambda - \Lambda_0 ) \biggl\{ \Lambda^3
-2\Lambda 
+\Lambda^2\cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  
 \biggr\}
+ \eta\cot(\eta-B)  \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr\}
</math>
  </td>
</tr>
</table>

Simplifying again produces,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Lambda^2 \biggl\{ 
1 
- \Lambda^3  \cot(\Lambda - E) 
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr\}
-  \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \Lambda^2 \cot(\Lambda-E) \biggl\{ \Lambda^3
-2\Lambda 
+\Lambda^2\cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  
 \biggr\}
+ \Lambda \cot(\Lambda-E)  \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E) 
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Lambda^3 \cot(\Lambda - E)  
- \Lambda^5  \cot(\Lambda - E) 
- \Lambda^5 \cot^3(\Lambda - E) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\Lambda^5 \cot(\Lambda-E) - 2\Lambda^3 \cot(\Lambda-E) 
+\Lambda^4\cot^2(\Lambda - E) 
+ \Lambda^5 \cot^3(\Lambda - E)  
+ \Lambda^3 \cot(\Lambda-E)   - \Lambda^4 \cot^2(\Lambda - E) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0\, . 
</math>
  </td>
</tr>
</table>

EXCELLENT !!  Keep trying to simplify the more general expression &hellip;


<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{ \Lambda^4(\Lambda - \Lambda_0 )^2}{2b_0 g_\mathrm{F}^2} \cdot 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \Lambda^2  
- \Lambda^5  \cot(\Lambda - E) 
- \Lambda^5 \cot^3(\Lambda - E) 
- \Lambda^2 + \Lambda^3 \cot(\Lambda - E)  + \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4 \biggr\}
- \Lambda_0 \biggl\{ \Lambda 
- \Lambda^4  \cot(\Lambda - E) 
- \Lambda^4 \cot^3(\Lambda - E) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \Lambda_0 (\Lambda - \Lambda_0 ) \biggl\{ 
(3-\Lambda^2)
- (\Lambda + \Lambda^3 ) \cot(\Lambda - E) 
- \Lambda^2 \cot^2(\Lambda - E)  
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \eta\cot(\eta-B)  (\Lambda ) \biggl\{ \Lambda^3
-2\Lambda 
+\Lambda^2\cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  
 \biggr\}
+ \eta\cot(\eta-B)  \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-  \eta\cot(\eta-B)  (\Lambda_0) \biggl\{ \Lambda^3
-2\Lambda 
+\Lambda^2\cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \Lambda^2  
- \Lambda^5  \cot(\Lambda - E) 
- \Lambda^5 \cot^3(\Lambda - E) 
- \Lambda^2 + \Lambda^3 \cot(\Lambda - E)  + \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \eta\cot(\eta-B) \biggl[ \Lambda^4
-2\Lambda^2 
+\Lambda^3 \cot(\Lambda - E) 
+ \Lambda^4 \cot^2(\Lambda - E)  
+ \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \Lambda_0 \biggl\{ (\Lambda - \Lambda_0 ) \biggl[ 
(3-\Lambda^2)
- (\Lambda + \Lambda^3 ) \cot(\Lambda - E) 
- \Lambda^2 \cot^2(\Lambda - E)  
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr]
+ \biggl[ \Lambda 
- \Lambda^4  \cot(\Lambda - E) 
- \Lambda^4 \cot^3(\Lambda - E) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \eta\cot(\eta-B) \biggl[ \Lambda^3
-2\Lambda 
+\Lambda^2\cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 
\biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4 
+ (\Lambda^3 - \Lambda^5)  \cot(\Lambda - E) 
- \Lambda^5 \cot^3(\Lambda - E) 
\biggr]
+  \eta\cot(\eta-B) \biggl[ \biggl(1- \frac{ a_0 }{b_0}\biggr)\Lambda^4
- \Lambda^2 
+ \Lambda^4 \cot^2(\Lambda - E)  
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \Lambda_0 \biggl\{ (\Lambda - \Lambda_0 ) \biggl[ 
(3-\Lambda^2)
- (\Lambda + \Lambda^3 ) \cot(\Lambda - E) 
- \Lambda^2 \cot^2(\Lambda - E)  
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr]
+ \biggl[ \Lambda 
- \Lambda^4  \cot(\Lambda - E) 
- \Lambda^4 \cot^3(\Lambda - E) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \eta\cot(\eta-B) \biggl[ \Lambda^3
-2\Lambda 
+\Lambda^2\cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>

----

Simplify again &hellip;

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 
\biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4 
+ (\Lambda^3 - \Lambda^5)  \cot(\Lambda - E) 
- \Lambda^5 \cot^3(\Lambda - E) 
\biggr]
+  \eta\cot(\eta-B) \biggl[ \biggl(1- \frac{ a_0 }{b_0}\biggr)\Lambda^4
- \Lambda^2 
+ \Lambda^4 \cot^2(\Lambda - E)  
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 
\biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4 
+ (\Lambda^3 - \Lambda^5)  \cot(\Lambda - E) 
- \Lambda^5 \cot^3(\Lambda - E) 
\biggr]
+   \biggl[
(\Lambda^5 
- \Lambda^3) \cot(\Lambda-E)
- \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^5\cot(\Lambda-E)
+ \Lambda^5 \cot^3(\Lambda - E)  
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{ a_0 }{b_0}\biggr) \biggl[ \Lambda^4 
- \Lambda^5\cot(\Lambda-E)\biggr] \, .
</math>
  </td>
</tr>
</table>

======First Argument Relationships Guess======

Let's try the relationship,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\eta - B)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~m(\Lambda - E) \, .</math>
  </td>
</tr>
</table>
Then, for example, if <math>~m = 2</math>, we can make the replacement, 
<div align="center">
<math>~\cot(\eta - B) = \cot[2(\Lambda - E)] = \frac{\cot^2(\Lambda - E) - 1}{2\cot(\Lambda - E)} \, .</math>
</div>
And, alternatively, if <math>~m = 3</math>, we can make the replacement, 
<div align="center">
<math>~\cot(\eta - B) = \cot[3(\Lambda - E)] = \frac{\cot^3(\Lambda - E) - 3\cot(\Lambda - E)}{3\cot^2(\Lambda - E)-1} \, .</math>
</div>


<table border="1" width="60%" align="center" cellpadding="8"><tr><td align="left">
<div align="center"><font color="red">'''IMPLICATIONS'''</font></div>

Given that we also are assuming the relationship,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{g_\mathrm{F}} (\Lambda - \Lambda_0) \, ,</math>
  </td>
</tr>
</table>
we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m\Lambda - \eta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~mE - B</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ m\Lambda - \frac{1}{g_\mathrm{F}} (\Lambda - \Lambda_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~mE - B</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \Lambda \biggl[m - \frac{1}{g_\mathrm{F}} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~mE - B -  \frac{\Lambda_0}{g_\mathrm{F}} \, . </math>
  </td>
</tr>
</table>
In order for this statement to be true for all <math>~\Lambda</math>, the RHS and the LHS must independently be zero.  Hence, we require,
<div align="center">
<math>~g_\mathrm{F} = \frac{1}{m}</math> &nbsp;  &nbsp; &nbsp; and  &nbsp; &nbsp; &nbsp;<math>~\Lambda_0 = E - \frac{B}{m} \, .</math>
</div>
</td></tr></table>

Let's try <math>~m=2</math>:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ 2\cot(\Lambda - E) \cdot \frac{ \Lambda^4(\Lambda - \Lambda_0 )^2}{2b_0 g_\mathrm{F}^2} \cdot 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\cot(\Lambda - E)\biggl[ 
\biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4 
+ (\Lambda^3 - \Lambda^5)  \cot(\Lambda - E) 
- \Lambda^5 \cot^3(\Lambda - E) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  2(\Lambda - \Lambda_0) [\cot^2(\Lambda - E) - 1]\biggl[ \biggl(1- \frac{ a_0 }{b_0}\biggr)\Lambda^4
- \Lambda^2 
+ \Lambda^4 \cot^2(\Lambda - E)  
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \Lambda_0 \biggl\{ 2(\Lambda - \Lambda_0 )\cot(\Lambda - E) \biggl[ 
(3-\Lambda^2)
- (\Lambda + \Lambda^3 ) \cot(\Lambda - E) 
- \Lambda^2 \cot^2(\Lambda - E)  
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  2(\Lambda - \Lambda_0)[\cot^2(\Lambda - E) - 1] \biggl[ \Lambda^3
-2\Lambda 
+\Lambda^2\cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2\cot(\Lambda - E)\biggl[ \Lambda 
- \Lambda^4  \cot(\Lambda - E) 
- \Lambda^4 \cot^3(\Lambda - E) 
\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~\cot(\Lambda - E) \cdot \frac{ \Lambda^4(\Lambda - \Lambda_0 )^2}{2b_0 g_\mathrm{F}^2} \cdot 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \Lambda_0 (\Lambda - \Lambda_0) \biggl[ \Lambda^3
-2\Lambda 
+\Lambda^2\cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-  (\Lambda - \Lambda_0) \biggl[ \biggl(1- \frac{ a_0 }{b_0}\biggr)\Lambda^4
- \Lambda^2 
+ \Lambda^4 \cot^2(\Lambda - E)  
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\cot(\Lambda - E)\biggl[ 
\biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4 
+ (\Lambda^3 - \Lambda^5)  \cot(\Lambda - E) 
- \Lambda^5 \cot^3(\Lambda - E) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\Lambda_0 (\Lambda - \Lambda_0 )\cot(\Lambda - E) \biggl[ 
(3-\Lambda^2)
- (\Lambda + \Lambda^3 ) \cot(\Lambda - E) 
- \Lambda^2 \cot^2(\Lambda - E)  
- \Lambda^3 \cot^3(\Lambda - E) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \Lambda_0 \cot(\Lambda - E)\biggl[ \Lambda 
- \Lambda^4  \cot(\Lambda - E) 
- \Lambda^4 \cot^3(\Lambda - E) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  (\Lambda - \Lambda_0) \cot^2(\Lambda - E) \biggl[ \biggl(1- \frac{ a_0 }{b_0}\biggr)\Lambda^4
- \Lambda^2 
+ \Lambda^4 \cot^2(\Lambda - E)  
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-  \Lambda_0 (\Lambda - \Lambda_0)\cot^2(\Lambda - E)  \biggl[ \Lambda^3
-2\Lambda 
+\Lambda^2\cot(\Lambda - E) 
+ \Lambda^3 \cot^2(\Lambda - E)  \biggr]
</math>
  </td>
</tr>
</table>

======Second Argument Relationships Guess======
Let's go back up to the general expression,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{ \Lambda^4(\Lambda - \Lambda_0 )^2}{2b_0 g_\mathrm{F}^2} \cdot 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\Lambda - \Lambda_0 ) \biggl\{ 
(3-\Lambda^2)(\Lambda - \Lambda_0 )
- (\Lambda + \Lambda^3 )(\Lambda - \Lambda_0 ) \cot(\Lambda - E) 
- \Lambda^2 (\Lambda - \Lambda_0 )\cot^2(\Lambda - E)  
- \Lambda^3 (\Lambda - \Lambda_0 )\cot^3(\Lambda - E) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\Lambda - \Lambda_0 ) \biggl\{ 2\Lambda^3
-4\Lambda  
+2\Lambda^2 \cot(\Lambda - E) 
+ 2\Lambda^3 \cot^2(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-Q \biggl\{ \Lambda^3(\Lambda - \Lambda_0 )
-2\Lambda  (\Lambda - \Lambda_0 )
+\Lambda^2 (\Lambda - \Lambda_0 )\cot(\Lambda - E) 
+ \Lambda^3 (\Lambda - \Lambda_0 )\cot^2(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- Q \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr\} \, ,
</math>
  </td>
</tr>
</table>
where,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1 - \eta \cot(\eta - B) \biggr] \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \biggl[\frac{(\Lambda - \Lambda_0)}{g_\mathrm{F}}\biggr] \cot[ (\Lambda - \Lambda_0)/g_\mathrm{F} - B] \, . </math>
  </td>
</tr>
</table>

Then, let's try setting, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cot[ (\Lambda - \Lambda_0)/g_\mathrm{F} - B]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan(\Lambda - E)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \cot\biggl\{ \frac{\pi}{2} - \biggl[ \frac{\pi}{2} + B - (\Lambda - \Lambda_0)/g_\mathrm{F} \biggr] \biggr\}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan(\Lambda - E)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\pi}{2} + B - \frac{ (\Lambda - \Lambda_0) }{g_\mathrm{F}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Lambda - E \, .</math>
  </td>
</tr>
</table>
This will only work for all <math>~\Lambda</math> if, <math>~g_\mathrm{F} = -1</math>; in which case,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\pi}{2} + B + (\Lambda - \Lambda_0)  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Lambda - E </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ E </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Lambda_0 - \frac{\pi}{2} - B  \, . </math>
  </td>
</tr>
</table>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + (\Lambda - \Lambda_0)\tan(\Lambda - E) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ Q \cot(\Lambda-E)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cot(\Lambda-E) + (\Lambda - \Lambda_0) \, .</math>
  </td>
</tr>
</table>

This means that,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cot(\Lambda-E) \cdot \frac{ \Lambda^4(\Lambda - \Lambda_0 )^2}{2b_0 g_\mathrm{F}^2} \cdot 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cot(\Lambda-E)(\Lambda - \Lambda_0 ) \biggl\{ 
(3-\Lambda^2)(\Lambda - \Lambda_0 )
- (\Lambda + \Lambda^3 )(\Lambda - \Lambda_0 ) \cot(\Lambda - E) 
- \Lambda^2 (\Lambda - \Lambda_0 )\cot^2(\Lambda - E)  
- \Lambda^3 (\Lambda - \Lambda_0 )\cot^3(\Lambda - E) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \cot(\Lambda-E)(\Lambda - \Lambda_0 ) \biggl\{ 2\Lambda^3
-4\Lambda  
+2\Lambda^2 \cot(\Lambda - E) 
+ 2\Lambda^3 \cot^2(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-[\cot(\Lambda-E) + (\Lambda - \Lambda_0)] \biggl\{ \Lambda^3(\Lambda - \Lambda_0 )
-2\Lambda  (\Lambda - \Lambda_0 )
+\Lambda^2 (\Lambda - \Lambda_0 )\cot(\Lambda - E) 
+ \Lambda^3 (\Lambda - \Lambda_0 )\cot^2(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- [\cot(\Lambda-E) + (\Lambda - \Lambda_0)]  \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\Lambda - \Lambda_0 ) \biggl\{ 
(3-\Lambda^2)(\Lambda - \Lambda_0 )\cot(\Lambda-E)
- (\Lambda + \Lambda^3 )(\Lambda - \Lambda_0 ) \cot^2(\Lambda - E) 
- \Lambda^2 (\Lambda - \Lambda_0 )\cot^3(\Lambda - E)  
- \Lambda^3 (\Lambda - \Lambda_0 )\cot^4(\Lambda - E) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\Lambda - \Lambda_0 ) \biggl\{ 2\Lambda^3\cot(\Lambda-E)
-4\Lambda  \cot(\Lambda-E)
+2\Lambda^2 \cot^2(\Lambda - E) 
+ 2\Lambda^3 \cot^3(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-(\Lambda - \Lambda_0) \biggl\{ \Lambda^3(\Lambda - \Lambda_0 )
-2\Lambda  (\Lambda - \Lambda_0 )
+\Lambda^2 (\Lambda - \Lambda_0 )\cot(\Lambda - E) 
+ \Lambda^3 (\Lambda - \Lambda_0 )\cot^2(\Lambda - E)  
 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- (\Lambda - \Lambda_0)  \biggl\{  \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
 \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-[\cot(\Lambda-E) ] \biggl\{ \Lambda^3(\Lambda - \Lambda_0 )
-2\Lambda  (\Lambda - \Lambda_0 )
+\Lambda^2 (\Lambda - \Lambda_0 )\cot(\Lambda - E) 
+ \Lambda^3 (\Lambda - \Lambda_0 )\cot^2(\Lambda - E)  
+ \Lambda^2 - \Lambda^3 \cot(\Lambda - E)  - \biggl( \frac{ a_0 }{b_0}\biggr)\Lambda^4
\biggr\} \, .
</math>
  </td>
</tr>
</table>