Question

$f\left(x\right)=\frac{2x}{1+x^2},$ then prove -$f\left(\tan \theta \right)=\sin 2\theta$

Answer 1

$f\left(x\right)=\frac{2x}{1+x^2}$

$f\left(\tan \theta \right)=\frac{2\tan \theta }{1+{\tan }^2\theta }$

$=\frac{2\frac{\sin \theta }{\cos \theta }}{{sec}^2\theta }$

$=\frac{2\sin \theta \times {\cos }^2\theta }{\cos \theta }$

$=2\sin \theta \cos \theta$

$=\sin 2\theta =R.H.S$