Question

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

Answer 1

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