Question

Prove the following,

$3co{s}^{-1}x=co{s}^{-1}(4x^3-3x),xϵ[\frac{1}{2},1]$.

Answer 1

Let $x=cos\theta$
LHS = $3co{s}^{-1}x=3co{s}^{-1}\left(cos\theta \right)=3\theta$
RHS = $co{s}^{-1}(4x^3-3x)=co{s}^{-1}[4co{s}^3\theta -3cos\theta ]=co{s}^{-1}\left[cos3\theta \right]=3\theta [\because cos3\theta =4co{s}^3\theta -3cos\theta ]$
$\therefore$ LHS=RHS

Hence, proved