Question:
$f(x)=\frac{2 x}{\left(1+x^{2}\right)}$ then show that $f(\tan \theta)=\sin 2 \theta$.
Solution:
Given: $f(x)=\frac{2 x}{\left(1+x^{2}\right)}$
Need to prove: f(tanθ) = sin 2θ
$f(\tan \theta)=\frac{2 \tan \theta}{1+\tan ^{2} \theta}$
$\Rightarrow f(\tan \theta)=\frac{2 \tan \theta}{\sec ^{2} \theta}\left[\operatorname{as} 1+\tan ^{2} \theta=\sec ^{2} \theta\right]$
$\Rightarrow f(\tan \theta)=2 \frac{\sin \theta}{\cos \theta} \cos ^{2} \theta_{\text {[as }} \tan \theta=\frac{\sin \theta}{\cos \theta}$ and $\sec \theta=\frac{1}{\cos \theta]}$
$\Rightarrow f(\tan \theta)=2 \sin \theta \cos \theta=\sin 2 \theta$ [Proved]