Prove the following:

    2. 3\cos^{-1} x = \cos^{-1}(4x^3 - 3x), \;\;x\in\left[\frac{1}{2},1 \right ]

Answers (1)

Given to prove 3\cos^{-1} x = \cos^{-1}(4x^3 - 3x), \;\;x\in\left[\frac{1}{2},1 \right ].

Take \cos^{-1}x = \theta  or \cos \theta = x;

Then we have;

R.H.S.

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

\cos^{-1}(4\cos^3 \theta - 3\cos\theta)      \left [ \because 4\cos^3 \theta - 3\cos\theta = \cos3 \theta \right ]

\cos^{-1}(\cos3\theta)

3\theta

3\cos^{-1}x = L.H.S

Hence Proved.

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