Question

If ${\sec }^{-1}\left(x\right)={\text{cosec}}^{-1}\left(y\right)$ then ${\cos }^{-1}\left(\frac{{\displaystyle 1}}{{\displaystyle x}}\right)+{\cos }^{-1}\left(\frac{{\displaystyle 1}}{{\displaystyle y}}\right)=$

• A. $\mathrm{\pi }$
• B. $\frac{\mathrm{\pi }}{4}$
• C. $\frac{-\mathrm{\pi }}{2}$
• D. $\frac{\mathrm{\pi }}{2}$

Answer 1

The correct option is D

$\frac{\mathrm{\pi }}{2}$

Simplification and calculation of the given expression:

We can write the expression as

$\because {\sec }^{-1}x={\mathrm{cosec}}^{-1}y$

$$\begin{gathered} {\cos }^{-1}\left(\frac{1}{x}\right)={\sin }^{-1}\left(\frac{1}{y}\right)\left[\because \sec \theta =\frac{1}{\cos \theta }\right] \\ {\cos }^{-1}\left(\frac{1}{x}\right)=\left(\frac{\mathrm{\pi }}{2}\right)-{\cos }^{-1}\left(\frac{1}{y}\right)\left[\because \sin (90°-\theta )=\cos \theta \right] \\ {\cos }^{-1}\left(\frac{1}{x}\right)+{\cos }^{-1}\left(\frac{1}{y}\right)=\frac{\mathrm{\pi }}{2} \end{gathered}$$

Hence , the correct option is (D).