Question

If $2{\tan }^{-1}(\cos x)={\tan }^{-1}(2cosecx),$ 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

Answer: (B)

We know that
$2{\tan }^{-1}x={\tan }^{-1}\frac{2x}{1-x^2}$
Hence: $2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(\frac{2\cos x}{1-{\cos }^{2}x}\right)$
$2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2cosecx\right)$
so, ${\tan }^{-1}\left(\frac{2\cos x}{1-{\cos }^{2}x}\right)={\tan }^{-1}\left(2cosecx\right)$
$\Rightarrow \frac{2\cos x}{{\sin }^{2}x}=2cosecx$
$\Rightarrow 2\cos x=2{\sin }^{2}xcosecx=2\sin x$ or $\tan x=1$
so, $x=\frac{\pi }{4}$ [The other value $\frac{3\pi }{4}$ is not valid since this lies in the third quadrant]