Question

If $2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2cosecx\right)$, then the value of x is.

• A. $\frac{3\pi }{4}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. None of these

Answer 1

The correct option is B $\frac{\pi }{4}$

$2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2cosecx\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{2cosx}{1-co{s}^{2}x}\right)={\tan }^{-1}\left(2cosecx\right)$ .... $[\because 2{\tan }^{-1}x={\tan }^{-1}(\frac{2x}{1-x^2}\left)\right]$

$\Rightarrow \frac{2\cos x}{1-{\cos }^{2}x}=2cosecx$

$\Rightarrow \frac{2\cos x}{{\sin }^{2}x}=2cosecx$

$\Rightarrow \sin x=\cos x\Rightarrow x=\frac{\pi }{4}$

Hence, option B is correct.