Question

The value of definite integral $\underset{-1}{\overset{1}{\int }}\left({\tan }^{-1}\left(\sin \left({\cos }^{-1}x\right)\right)+{\cot }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)\right)dx$ 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, $\underset{-1}{\overset{1}{\int }}\left({\tan }^{-1}\theta +{\cot }^{-1}\theta \right)d\theta$, where $\theta =\cos \left({\sin }^{-1}x\right)=\frac{\pi }{2}(2)=\pi$