trigonometry - How to prove that $csc x - 2cos x cot 2x = 2 sin x$

My idea was...

\begin{align}
& \text{LHS} \\[10pt]
= {} & \frac 1 {\sin x} - \frac{2\cos x}{\tan2x} \\[10pt]

= {} & \frac{\tan2x-\sin2x}{\sin x\tan2x}
\end{align}

from here, I don't know how to continue, please help! thanks

ps, please teach me how to use the "divide" symbol

Answer

Somewhere you'll need a double-angle formula:
$$
\tan(2x) = \frac{2\tan x}{1 - \tan^2 x}

$$
or
$$
\tan(2x) = \frac{\sin(2x)}{\cos(2x)} = \frac{2\sin x\cos x}{\cos^2 x - \sin^2 x}.
$$