Question

If ${\sin }^{-1}x+{\cos }^{-1}y=\frac{2\pi }{5}$, then ${\cos }^{-1}x+{\sin }^{-1}y$ is

• A. $\frac{2\pi }{5}$
• B. $\frac{3\pi }{5}$
• C. $\frac{4\pi }{5}$
• D. $\frac{3\pi }{10}$

Answer 1

The correct option is B $\frac{3\pi }{5}$

$\text{Let}{\cos }^{-1}x+{\sin }^{-1}y=t$
$\text{Then},{\sin }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}x+{\sin }^{-1}y=\frac{2\pi }{5}+t$
$\Rightarrow ({\sin }^{-1}x+{\cos }^{-1}x)+({\sin }^{-1}y+{\cos }^{-1}y)=\frac{2\pi }{5}+t$
$\Rightarrow \frac{\pi }{2}+\frac{\pi }{2}=\frac{2\pi }{5}+t$
$\Rightarrow t=\frac{3\pi }{5}$