Question

How do prove this identity? I’m so lost on trig identities…

Answer 1

Given:

`(2\tan x)/(1+\tan^2x)=\sin 2x`

Take the left-hand side of the equation,

`\begin{gathered} \text{LHS}=(2\tan x)/(1+\tan^2x) \\ \text{Use the identity: 1+tan}^2x=sex^2x \\ =(2\tan x)/(\sec^2x) \\ =(2(\sin x)/(\cos x))/((1)/(\cos^2x))\ldots.\ldots\text{.. Since tanx=}(sinx)/(\cos x),\text{secx}=(1)/(\cos x) \\ =2(\sin x)/(\cos x)*\cos ^2x \\ =2\sin x*\cos x \\ =\sin 2x\ldots\ldots...\ldots\text{ Since }\sin 2\text{x=2}\sin x\cdot\cos x \\ =\text{ Left hand side} \end{gathered}`

Hence, it is proved that,

`(2\tan x)/(1+\tan^2x)=\sin 2x`