Question

Prove: $2{\sin }^{-1}x={\cos }^{-1}(1-2x^2).$

Answer 1

To prove:-

$2{\sin }^{-1}x={\cos }^{-1}(1-2x^2)$

Proof:-

$2{\sin }^{-1}x={\cos }^{-1}(1-2x^2)⟶\left(1\right)$

Let $x=\sin \theta$

Substituting the value of x in ${eq}^n\left(1\right)$, we have

$2{\sin }^{-1}\left(\sin \theta \right)={\cos }^{-1}(1-2{\sin }^2\theta )$

$\Rightarrow 2\theta =co{s}^{-1}\left(\cos \left(2\theta \right)\right)\{\because \cos 2\theta =1-2{\sin }^2\theta \}$

$\Rightarrow 2\theta =2\theta$

$L.H.S.=R.H.S.$

Hence proved.