Prove that: 3 ^ - 1 x = ^ - 1 ( 4 x^3 - 3x ),x [ 1 2 ,1 ] - Vidyadip

Question

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

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Answer

We know that: $\cos 3\theta = 4 {\cos ^3}\theta - 3\cos \theta$
Put $\cos \theta = x$
$\Rightarrow \theta = {\cos ^{ - 1}}x$
$\therefore \cos 3\theta = 4{x^3} - 3x$
$ \Rightarrow 3\theta = {\cos ^{ - 1}}\left( {3x - 4{x^3}} \right)$
Putting $\theta = {\cos ^{ - 1}}x$,
$3\cos^{-1}x = \cos^{-1}(4x^3 - 3x)$ Hence Proved.

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