Question

Differentiate ${\tan }^{-1}\left(\frac{2^{x+1}}{1-4^x}\right)$ w.r.t. $x.$

Answer 1

$\frac{2\tan \theta }{1+{\tan }^2\theta }$
$=\frac{2\tan \theta }{{\sec }^2\theta }$
$=2\tan \theta \times {\cos }^2\theta$
$=2\frac{\sin \theta }{\cos \theta }\times {\cos }^2\theta$
$=2\sin \theta \cos \theta$
$=\sin 2\theta$
${\cos }^{-1}\left(\frac{2\tan \theta }{1+{\tan }^2\theta }\right)={\cos }^{-1}\left(\sin 2\theta \right)$
$={\cos }^{-1}\left[\cos \right(\pi /2-2\theta \left)\right]$
$=\pi /2-2\theta$