If tan^{- 1} (x - 1)/(x - 2) + tan^{- 1} (x + 1)/(x - 1) = π/4, find the value of x

Solution:

Inverse trigonometric functions as a topic of learning are closely related to the basic trigonometric ratios.

The domain (θ  value)  and the range(answer) of the trigonometric ratio are changed to the range and domain of the inverse trigonometric function.

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

It is given that

tan^{- 1} (x - 1)/(x - 2) + tan^{- 1} (x + 1)/(x - 1) = π / 4

Since tan^{- 1} x + tan^{- 1} y = tan^{- 1} (x + y)/(1 - xy)

Therefore,

⇒ tan^{- 1} [(x - 1)/(x - 2) + (x + 1)/(x + 2)] / [1 - (x - 1)/(x - 2)(x + 1) / (x + 2)] = π / 4

⇒ tan^{- 1} [(x - 1)(x + 2) + (x + 1)(x - 2)] / [(x + 2) - (x - 1)(x + 1) / (x + 1)] = π / 4

⇒ tan^{- 1} [(x² + x - 2 + x² - x - 2)/(x² - 4 - x² + 1)] = π / 4

⇒ tan^{- 1} [(2x² - 4)/(- 3)] = π / 4

⇒ tan [tan^{- 1} (4 - 2x²) / 3]

= tan π / 4

⇒ (4 - 2x²) / 3 = 1

⇒ 4 - 2x² = 3

⇒ 2x² = 1

⇒ x² = 1/2

⇒ x = ± 1/√2

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NCERT Solutions for Class 12 Maths - Chapter 2 Exercise 2.2 Question 15

If tan^{- 1} (x - 1)/(x - 2) + tan^{- 1} (x + 1)/(x - 1) = π/4, find the value of x

Summary:

Given that tan^{- 1} (x - 1)/(x - 2) + tan^{- 1} (x + 1)/(x - 1) = π/4, the value of x is ± 1/√2