Question

If $f\left(x\right)=\frac{2x}{1+x^2}$, show that $f\left(\tan \theta \right)=\sin 2\theta$.

Answer 1

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

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

$\therefore f\left(\tan \theta \right)=2\frac{\sin \theta }{\cos \theta }\times {\cos }^2\theta$

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

$f\left(\tan \theta \right)=\sin 2\theta$