Question

If $\sin ({\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x)=1$, then find the value of x.

Answer 1

$\sin ({\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x)=1$

${\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x={\sin }^{-1}1$

${\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x={\sin }^{-1}\left(\sin \frac{\pi }{2}\right)$

${\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x=\frac{\pi }{2}$

But, we know, ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$

Therefore,

${\sin }^{-1}\frac{1}{5}={\sin }^{-1}x$

$x=\frac{1}{5}$