Question

Find the value of x which satisfy equation : $2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(\text{2 csc x}\right)$

• A. $x=n\pi +\frac{\pi }{4}$
• B. $x=n\pi -\frac{\pi }{4}$
• C. $x=n\pi +\frac{\pi }{2}$
• D. $x=n\pi -\frac{\pi }{2}$

Answer 1

The correct option is A $x=n\pi +\frac{\pi }{4}$

Given, $2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2\csc x\right)$
$\Rightarrow {\tan }^{-1}\left(\frac{2\cos x}{1-{\cos }^{2}x}\right)={\tan }^{-1}\left(2\csc x\right)$
$\Rightarrow \frac{2\cos x}{{\sin }^{2}x}=2\csc x\Rightarrow \frac{2\cos x}{{\sin }^{2}x}=\frac{2}{\sin x}$
$\Rightarrow \cos x=\sin x\Rightarrow x=n\pi +\frac{\pi }{4}$