# How do you verify the identity 4 csc 2x = 2 csc^2x tan x?

Apr 28, 2015

We will be using a Trigonometric identity to prove this-
color(blue)(sin(2x) = 2sinxcosx

The left hand side is $4 \csc 2 x$

$= \frac{4}{\sin 2 x}$ (Because $\csc x = \frac{1}{\sin} x$)

$= \frac{4}{2 \sin x \cos x}$

 = color(green)(2/(sinxcosx)

The right hand side is $2 {\csc}^{2} x \tan x$

$= 2 \cdot \left(\frac{1}{\sin} ^ 2 x\right) \cdot \left(\sin \frac{x}{\cos} x\right)$ (Because $\tan x = \sin \frac{x}{\cos} x$)

 = color(green)(2/(sinxcosx)

Hence proved.