Question

The value of $\sin ^{-1}(\cos 2)-\cos ^{-1}(\sin 2)+\tan ^{-1}(\cot 4)-\cot ^{-1}(\tan 4)+\sec ^{-1}(\operatorname{cosec} 6)-\operatorname{cosec}^{-1}(\sec 6)$ is

• A. 0
• B. $3 \pi$
• C. $8-3 \pi$
• D. $5 \pi-16$

Answer 1

Answer: (D)

Given expression $=\left\{\frac{\pi}{2}-\cos ^{-1}(\cos 2)\right\}-\left\{\frac{\pi}{2}-\sin ^{-1}(\sin 2)\right\}$
$+\left\{\frac{\pi}{2}-\cot ^{-1}(\cot 4)\right\}-\left\{\frac{\pi}{2}-\tan ^{-1}(\tan 4)\right\}+\left\{\frac{\pi}{2}-\operatorname{cosec}^{-1}(\operatorname{cosec} 6)\right\}-\left\{\frac{\pi}{2}-\sec ^{-1}(\sec 6)\right\} $
$=\sin ^{-1}(\sin 2)-\cos ^{-1}(\cos 2)+\tan ^{-1}(\tan 4)-\cot ^{-1}(\cot 4)+\sec ^{-1}(\sec 6)-\operatorname{cosec}^{-1}(\operatorname{cosec} 6) $
$=(\pi-2)-2+(4-\pi)-(4-\pi)+(2 \pi-6)-(6-2 \pi)=\pi-4+4 \pi-12=5 \pi-16$