Question

The value of $\sin \left({\sin }^{-1}\frac{1}{3}+{\sec }^{-1}3\right)+$ $\cos \left({\tan }^{-1}\frac{1}{2}+{\tan }^{-1}2\right)$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (A)

Key Idea: ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$ and ${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}$ $\sin \left({\sin }^{-1}\frac{1}{3}+{\sec }^{-1}3\right)$ $+\cos \left({\tan }^{-1}\frac{1}{2}+{\tan }^{-1}2\right)$ $=\sin \left({\sin }^{-1}\frac{1}{3}+{\cos }^{-1}\frac{1}{3}\right)$ $+\cos \left({\tan }^{-1}\frac{1}{2}+{\cot }^{-1}\frac{1}{2}\right)$ $=\sin \left(\frac{\pi }{2}\right)+\cos \left(\frac{\pi }{2}\right)=1$