Question

If the value of $sin\left({\left(cos\right)}^{-1}x\right)$ is $\sqrt{1-x^n}$ then find value of $n$ .

Answer 1

Answer: 2

We have, $\sin \left({\cos }^{-1}x\right)$

$\therefore {\cos }^{-1}x={\sin }^{-1}\sqrt{1-x^2}$
Then, $\sin \left({\cos }^{-1}x\right)=\sin \left({\sin }^{-1}\sqrt{1-x^2}\right)$
$\Rightarrow \sin \left({\cos }^{-1}x\right)=\sqrt{1-x^2}\dots (i)\left(\because \sin \left({\sin }^{-1}x\right)=x,x\in [-1,1]\right)$
Given, $\sin \left({\cos }^{-1}x\right)=\sqrt{1-x^n}\dots$ (ii)
Compare (i) \& (ii) we get, $n=2$.