Question

The value of ${\sin }^{-1}\left(\sin 4\right)+{\cos }^{-1}\left(\cos 5\right)+{\tan }^{-1}\left(\tan 6\right)+{\cot }^{-1}\left(\cot 7\right)$ is

• A. $13-2\pi$
• B. $9-\pi$
• C. $9+2\pi$
• D. $4-\pi$

Answer 1

The correct option is D $4-\pi$

We know, ${\sin }^{-1}\left(\sin x\right)=\pi -x,\frac{\pi }{2}\le x\le \frac{3\pi }{2}$,
${\cos }^{-1}\left(\cos x\right)=2\pi -x,\pi \le x\le 2\pi$,
${\tan }^{-1}\left(\tan x\right)=x-2\pi ,\frac{3\pi }{2}\le x\le \frac{5\pi }{2}$,
${\cot }^{-1}\left(\cot x\right)=x-2\pi ,2\pi <x<3\pi$
$\Rightarrow {\sin }^{-1}\left(\sin 4\right)+{\cos }^{-1}\left(\cos 5\right)+{\tan }^{-1}\left(\tan 6\right)+{\cot }^{-1}\left(\cot 7\right)$
$=\pi -4+2\pi -5+6-2\pi +7-2\pi$
$=4-\pi$