Question

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

Answer 1

Given: $\sin ({\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x)=1$
Taking $1=\sin \frac{\pi }{2}$
$\Rightarrow \sin ({\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x)=\sin \frac{\pi }{2}$
Comparing angles
$\Rightarrow {\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x=\frac{\pi }{2}$
$\Rightarrow {\sin }^{-1}\frac{1}{5}=\frac{\pi }{2}-{\cos }^{-1}x$
$\because {\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$
$\Rightarrow {\sin }^{-1}\frac{1}{5}={\sin }^{-1}x$
$\Rightarrow x=\frac{1}{5}$
Hence, the value of $x$ is $\frac{1}{5}.$