# If $2\tan^{-1}(\cos x)=\tan^{-1}(2cosecx)$,then what is the value of $x$?

$\begin{array}{1 1} \pi \\ \ \frac{\pi}{2} \\ \frac{\pi}{4} \\ \frac{-\pi}{3} \end{array}$

Toolbox:
• $$2tan^{-1}y=tan^{-1}\large\frac{2y}{1-y^2}\:\:|y|<1$$
• $$1-cos^2x=sin^2x$$
By taking y=cosx, we get $$\large\frac{2y}{1-y^2}=\large\frac{2cosx}{1-cos^2x}=\large\frac{2cosx}{sin^2x}$$
$$\Rightarrow\: 2tan^{-1}cosx=tan^{-1}\large\frac{2cosx}{sin^2x}$$
$$tan^{-1} \bigg( \large\frac{2cosx}{1-cos^2x} \bigg) = tan^{-1}(2cosecx)$$
$$\Rightarrow \large\frac{2cosx}{sin^2x}=2\: cosecx$$
$$cosx=sin^2x.cosecx=sin^2x.\large\frac{1}{sinx}$$
$$\Rightarrow cosxsinx=sin^2x$$

$$\Rightarrow cosx=sinx$$

$$\Rightarrow x=\large\frac{\pi}{4}$$

edited Mar 15, 2013