Question

The value of definite integral $\int\limits_{-1}^1\left(\tan ^{-1}\left(\sin \left(\cos ^{-1} x\right)\right)+\cot ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)\right) d x$ is equal to

• A. 0
• B. -1
• C. 1
• D. $\pi$

Answer 1

Answer: (D)

As $\sin \left(\cos ^{-1} x\right)=\sin \left(\frac{\pi}{2}-\sin ^{-1} x\right)=\cos \left(\sin ^{-1} x\right)$
Hence, $\int\limits_{-1}^1\left(\tan ^{-1} \theta+\cot ^{-1} \theta\right) d \theta$, where $\theta=\cos \left(\sin ^{-1} x\right)=\frac{\pi}{2}(2)=\pi$