Question

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

Answer: (B)

$\frac{\pi }{4}$

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

$\Rightarrow \frac{2\cos x}{1-{\cos }^{2}x}=\frac{1}{\sin x}\Rightarrow 2\cos x\sin x=1-{\cos }^{2}x\Rightarrow 2\cos x\sin x={\sin }^{2}x\Rightarrow \sin x(2\cos x-\sin x)=0\Rightarrow \sin x=0,2\cos x-\sin x=0\Rightarrow \sin x=0,\cos x=sinx\Rightarrow x=n\pi ,\frac{\pi }{4}$

None of the options match with our general solution

So, option D is correct.