I have an equation:
$-\frac{\cos(kL_1)}{\sin(kL_1)} + 2\frac{1-\cos(kL_2)}{\sin(kL_2)} =0 $
that I would like to factor to separate my variables as such: $[f(kL_2)][f(kL_1)] =0$, I've been trying for a while using various trig identities and exponential forms, but I can't quite manage to separate them. Can anyone provide any help or suggest a way I might be able to do it?
*Edit 1: I know the functions can be separated such that: $[f(kL_2)][f(kL_1)] =1$ , but I worry as to how that will impact my analysis of the functions.
Here's the best I can do.
To make it easier, let $kL_1=a $ and $kL_2=b $. This becomes $\frac{\cos(a)}{\sin(a)} = 2\frac{1-\cos(b)}{\sin(b)} $ or $\cot(a) =2(\csc(b)-\cot(b)) =2\tan(b/2) $.