Question

The value of ${\sin }^{-1}x+{\sin }^{-1}\frac{1}{x}+{\cos }^{-1}x+{\cos }^{-1}\frac{1}{x}$ where ever defined is

• A. $\begin{array}{c}\pi \end{array}$
• B. $\begin{array}{c}\frac{\pi }{2}\end{array}$
• C. $\begin{array}{c}\frac{3\pi }{2}\end{array}$
• D. $2\pi$

Answer 1

The correct option is A $\begin{array}{c}\pi \end{array}$

The domain of ${\sin }^{-1}x+{\sin }^{-1}\frac{1}{x}+{\cos }^{-1}x+{\cos }^{-1}\frac{1}{x}$ is $\{-1,1\}$

${\sin }^{-1}x+{\sin }^{-1}\frac{1}{x}+{\cos }^{-1}x+{\cos }^{-1}\frac{1}{x}=\left[{\sin }^{-1}\right(x)+{\cos }^{-1}(x\left)\right]+[{\sin }^{-1}\left(\frac{1}{x}\right)+{\cos }^{-1}\left(\frac{1}{x}\right)]=\frac{\pi }{2}+\frac{\pi }{2}=\pi$