Prove 3 cos^{- 1} x = cos^{- 1} (4x³ - 3x), x ∈ [1/2, 1]

Solution:

Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios. 

Here the basic trigonometric function of Sin θ = y can be changed to θ = sin^-1 y

Let x = cos θ

Hence, cos^{- 1} (x) = θ

Now,

RHS = cos^{- 1} (4x³ - 3x)

Using trigonometric identity 

cos 3 θ =  4 cos³ θ - 3cosθ

= cos^{- 1} (4 cos³ θ - 3cosθ)

= cos^{- 1} (cos 3θ)

= 3 θ

On substituting the value of θ, we get

RHS = 3 cos^{- 1} x

= LHS

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NCERT Solutions for Class 12 Maths - Chapter 2 Exercise 2.2 Question 2

Prove 3 cos^{- 1} x = cos^{- 1} (4x³ - 3x), x ∈ [1/2, 1]

Summary:

Here we have proved that 3 cos^{- 1} x = cos^{- 1} (4x³ - 3x), x ∈ [1/2, 1]