Write the function in the simplest form : tan^{- 1} x/√(a² + x²), |x| < a

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 = a sin θ

⇒ θ = sin^{- 1} (x / a)

Hence,

tan^{- 1} x/√(a² + x²)

Using trigonometric identity 

= tan^{- 1}(a sinθ / √(a² - a² sin² θ)

= tan^{- 1} (a sin θ / a √(1 - sin² θ)

= tan^{- 1} (a sinθ / a cos θ)

= tan^{- 1} (tan θ)

= θ

= sin^{- 1} (x / a)

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NCERT Solutions for Class 12 Maths - Chapter 2 Exercise 2.2 Question 9

Write the function in the simplest form : tan^{- 1} x/√(a² + x²), |x| < a

Summary:

The function in the simplest form : tan^{- 1} x/√(a² + x²), |x| < a can be written as sin^{- 1} (x / a).  Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios