Question

Prove: $\frac{\sin x-\sin 3x+\sin 5x-\sin 7x}{\cos x-\cos 3x-\cos 5x+\cos 7x}=\cot 2x$

Answer 1

$\frac{sinx-sin3x+sin5x-sin7x}{cosx-cos3x-cos5x+cos7x}$

$=\frac{(sinx-sin3x)+(sin5x-sin7x)}{(cosx-cos3x)-(cosx-cos7x)}$

$=\frac{2cos2xsin(-x)+2cos6xsin(-x)}{2sin2xsinx-2sin6xsinx}$

$\frac{-2sin\left(x\right)[cos2+cos6x]}{2sinx[sin2x-sin6x]}$

$\frac{-2cos4\left(x\right)cos2x}{2cos4\left(x\right)sin(-2x)}$

=cot(2x)