Question

${\left(\frac{\cos A+\cos B}{\sin A-\sin B}\right)}^n+{\left(\frac{\sin A+\sin B}{\cos A-\cos B}\right)}^n=2{\cot }^n\left(\frac{A-B}{2}\right)$ or $0$ according as $n$ is even or odd positive integer.

Answer 1

LHS ${\left[\frac{2\cos \left\{(A+B)/2\right\}\cos \left\{(A-B)/2\right\}}{2\sin \left\{(A-B)/2\right\}\cos \left\{(A+B)/2\right\}}\right]}^n+{\left[\frac{2\sin \left\{(A+B)/2\right\}\cos \left\{(A-B)/2\right\}}{2\sin \left\{(A+B)/2\right\}\sin \left\{(B-A)/2\right\}}\right]}^n$

$={\cot }^n\left\{(A-B)/2\right\}+{\left[\cot \left\{(A-B)/2\right\}\right]}^n$

$=2{\cot }^n\left\{(A-B)/2\right\}$ or $0$
according as $n$ is even or odd as
${(-1)}^n=1$ or $-1$