Question

If ${\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)=\theta$, then the value of $cosec{}^{-1}(\sqrt{5})$ is

• A. $\left(\frac{\pi }{2}\right)+\theta$
• B. $\left(\frac{\pi }{2}\right)-\theta$
• C. $\frac{\pi }{2}$
• D. $-\theta$

Answer 1

Answer: (B)

Let, ${\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)=\theta$
$\Rightarrow \cos \theta =\frac{1}{\sqrt{5}}$
$\Rightarrow \sec \theta =\sqrt{5}$
$\Rightarrow {\sec }^{-1}(\sqrt{5})=\theta$
$\Rightarrow \frac{\pi }{2}-cosec{}^{-1}(\sqrt{5})=\theta$
$\Rightarrow cosec{}^{-1}(\sqrt{5})=\frac{\pi }{2}-\theta$