Question

If ${\sin }^{-1}x\in (0,\frac{\pi }{2})$, then the value of $\tan \left(\frac{{\cos }^{-1}\left(\sin \right({\cos }^{-1}x\left)\right)+\sin \left(\cos \right({\sin }^{-1}x\left)\right)}{2}\right)$ is:

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A $1$

$\text{solution:}\tan \left(\frac{{\cos }^{-1}\left(\sin \left({\cos }^{-1}x\right)\right)+{\sin }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)}{2}\right)$

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

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

$=\tan \left(\frac{\pi }{4}\right)=1$

$\text{Answer: option (A)}$