Question

If $\sin \left({\cos }^{-1}\frac{5}{13}+{\sin }^{-1}x\right)=1,$ find the value of x.

Answer 1

We have,

$\sin ({\cos }^{-1}\frac{5}{13}+{\sin }^{-1}x)=1$

${\cos }^{-1}\frac{5}{13}+{\sin }^{-1}x={\sin }^{-1}1$

${\cos }^{-1}\frac{5}{13}+{\sin }^{-1}x=\frac{\pi }{2}$

${\cos }^{-1}\frac{5}{13}=\frac{\pi }{2}-{\sin }^{-1}x$

We know that

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

Therefore,

${\cos }^{-1}\frac{5}{13}={\cos }^{-1}x$

$x=\frac{5}{13}$

Hence, this is the answer.