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

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