Prove that 3cos−1x=cos−1(4x3−3x),xϵ[21,1].

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Prove that 3{}^{-1}x = {}^{-1}(4x^3-3x), x {bmatrix}{1}{2}, 1{

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Prove that $3 cos^{- 1} x = cos^{- 1} (4 x^{3} - 3 x), x ϵ [\frac{1}{2}, 1]$ .

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Prove that

3 cos^{- 1} x = cos^{- 1} (4 x^{3} - 3 x), x ϵ [\frac{1}{2}, 1]

11 mins ago

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11 mins ago

We have to prove that

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

First consider RHS,

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

Let us take

x = cos θ

, we know that

cos (3 θ) = 4 co s^{3} θ - 3 cos θ

⟹ cos^{- 1} (4 x^{3} - 3 x) = cos^{- 1} (4 cos^{3} θ - 3 cos θ) = cos^{- 1} (cos (3 θ)) = 3 θ

∵ θ = co s^{- 1} x

⟹ cos^{- 1} (4 x^{3} - 3 x) = 3 co s^{- 1} x

∴ L H S = R H S

0 \leq 3 co s^{- 1} x \leq π

0 \leq cos^{1} x \leq π /3

∴ x ϵ [\frac{1}{2}, 1]

Hence proved.

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