Question

If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{2}$, then the value of ${\cos }^{-1}x+{\cos }^{-1}y=$

• A. $\frac{\mathrm{\pi }}{2}$
• B. $\frac{\mathrm{\pi }}{4}$
• C. $\mathrm{\pi }$
• D. $\frac{3\mathrm{\pi }}{4}$

Answer 1

The correct option is A

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

Explanation for the correct option.

Given that, ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{2}$
Recall: ${\sin }^{-1}x+{\cos }^{-1}y=\frac{\pi }{2}$
$\begin{array}{rcl}\therefore {\cos }^{-1}x+{\cos }^{-1}y& =& \left(\frac{\pi }{2}-{\sin }^{-1}x\right)+\left(\frac{\pi }{2}-{\sin }^{-1}y\right)\\ & =& \pi -\left({\sin }^{-1}x+{\sin }^{-1}y\right)\\ & =& \pi -\frac{\pi }{2}\\ & =& \frac{\pi }{2}\end{array}$
$\Rightarrow {\cos }^{-1}x+{\cos }^{-1}y=\frac{\pi }{2}$

Hence, option A is correct.