Question

Trig question? Or maybe using Euler's Identity? I think...solve for x
cot(x) - csc(x) = (c-a)/b

Answer 1

So, kept trying to find another way around this problem without getting to this point, but I am now quite convinced I have to solve it this way. a, b, and c are all constant terms. And thanks to wolframalpha I believe cot(x) - csc(x) = -tan(x/2). Only problem is I can't figure out how to show this...

Answer 2

\[\cot(x)-\csc(x)=\frac{\cos(x)}{\sin(x)}-\frac{1}{\sin(x)}=\frac{\cos(x)-1}{\sin(x)}=-\tan(\frac{x}{2})\]

I don't really like this though...
I think I would do this question a much more fun way :P
Let y=cot(x) so

\[\text{ so } \cot(x)-\csc(x)=y-\sqrt{1+y^2} \\ y-\sqrt{1+y^2}=\frac{c-a}{b} \\ \frac{c-a}{b}-y=-\sqrt{1+y^2} \\ (\frac{c-a}{b}-y)^2=1+y^2 \\ (\frac{c-a}{b})^2-2y \frac{c-a}{b}+y^2=1+y^2 \\ (\frac{c-a}{b})^2-2y \frac{c-a}{b}=1 \]
solve for y
then replace y with cot(x)