Prove that tan^{- 1} √x = 1/2 cos^{- 1} [(1 - x)/(1 + x)], x ∈ |0, 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 = tan² θ

Then,

√x = tan θ

θ = tan^-1√x

Therefore,

(1 - x) / (1 + x) = (1 - tan² θ)/(1 + tan² θ)

Thus,

RHS = 1/2 cos^{- 1} [(1 - x)/(1 + x)]

= 1/2 cos^{- 1} (cos 2θ)

= 1/2 × θ

= tan^-1√x

= LHS

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NCERT Solutions for Class 12 Maths - Chapter 2 Exercise ME Question 9

Prove that tan^{- 1} √x = 1/2 cos^{- 1} [(1 - x)/(1 + x)], x ∈ |0, 1|

Summary:

Hence we have proved by using inverse trigonometric functions that tan^{- 1} √x = 1/2 cos^{- 1} [(1 - x)/(1 + x)], x ∈ |0, 1|