Write the function in the simplest form : tan^-1[(√1 + x²) - 1]/x, x ≠ 0

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 = tan θ

⇒ θ = tan^{- 1} x

Hence,

tan^{- 1} [(√1 + x²) - 1] / x

= tan^{- 1} (√1 + tan ²θ) - 1) / tan θ

= tan^{- 1} (secθ - 1) / tanθ

= tan^{- 1} (1 - cosθ) / sinθ

= tan^{- 1} (2 sin² θ / 2) / (2 sinθ / 2 cos θ)

= tan^{- 1} (tanθ / 2)

= θ / 2

= 1/2 tan^{- 1} x

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NCERT Solutions for Class 12 Maths - Chapter 2 Exercise 2.2 Question 5

Write the function in the simplest form : tan^-1[(√1 + x²) - 1]/x, x ≠ 0

Summary:

The function in the simplest form : tan^-1[(√1 + x²) - 1]/x, x ≠ 0 can be expressed as 1/2 tan^{- 1} x