Question

Prove that $\sin 2x=\frac{2\tan x}{2+{\tan }^{2}x}$

Answer 1

$\sin 2x=\frac{2\tan x}{1+{\tan }^{2}x}$

$R.H.S$

$\frac{2\tan x}{1+{\tan }^{2}x}$

$=\frac{2\frac{\sin x}{\cos x}}{1+\frac{{\sin }^{2}x}{{\cos }^{2}x}}$

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

$=2\sin x.\cos x$

$=\sin 2x$

$=L.H.S$

$so,\sin 2x=\frac{2\tan x}{1+{\tan }^{2}x}$ proved.