Verifying Identities

Question

How can we verify the following identity?

1. $-\frac{\sin 2x+\sin x}{\cos 2x-\cos x} = \cot(\frac{x}{2})$

These are the double angle identities that can be applied:

$\sin 2x = 2\sin x\cos x$

$\cos 2x=\cos^2 x-\sin^2 x=1-2\sin^2 x=2\cos^2 x-1$

$\tan 2x = \frac{2\tan x}{1-\tan^2 x}$

Answer 1

Use $\sin2x=2\sin x\cos x$ and $\cos2x=2\cos^2x-1$ and cancel $\cos2x+1$ top and bottom to get it down to proving $${\sin x\over1-\cos x}=\cot(x/2)$$ Then use $\sin x=2\sin(x/2)\cos(x/2)$ and $\cos x=1-2\sin^2(x/2)$.