Question

Prove that $3{\cos }^{-1}x={\cos }^{-1}\left(4x^3-3x\right)x\in \left[1/2,1\right]$

Answer 1

To prove:-$3{\cos }^{-1}x={\cos }^{-1}(4x^3-3x),x\in [\frac{1}{2},1].$

Let ${\cos }^{-1}x=\theta ,thenx=\cos \theta .........\left(1\right)$

Proof:-

$RHS={\cos }^{-1}(4x^3-3x)$

$={\cos }^{-1}(4{\cos }^3\theta -3\cos \theta )$

$={\cos }^{-1}\left(\cos 3\theta \right)$

$=3\theta =3{\cos }^{-1}xbyequation\left(1\right)$

$LHS$

LHS=RHS

Hence proof