Question

Which of the following is/are true?
(i) $\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}, x \in[-1,1]$
(ii) $\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}, x \in R$
(iii) $\operatorname{cosec}^{-1} x+\sec ^{-1} x=\frac{\pi}{2},|x| \geq 1$

• A. (i), (ii)
• B. (ii), (iii)
• C. (i), (iii)
• D. (i), (ii), (iii)

Answer 1

Answer: (D)

(i) $\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}, x \in[-1,1]$
(ii) $\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}, x \in R$
(iii) $\operatorname{cosec}^{-1} x+\sec ^{-1} x=\frac{\pi}{2},|x| \geq 1$
Let $\sin ^{-1} x=y$. Then, $x=\sin y=\cos \left(\frac{\pi}{2}-y\right)$
Therefore, $\cos ^{-1} x=\frac{\pi}{2}-y=\frac{\pi}{2}-\sin ^{-1} x$
Hence, $ \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}$
Similarly, we can prove the other parts.