Write the function in the simplest form : tan^{- 1} (3a² x - x³)/(a³ - 3ax²), a > 0; - a/√3 ≤ x ≤ a/√3

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 tan θ

⇒ θ = tan^-1 (x/a)

tan^{- 1} (3a² x - x³)/(a³ - 3ax²)

Using trigonometric identity 

= tan^{- 1} (3a² × a tan θ - a³ tan³ θ) / (a³ - 3a.a² tan² θ)

= tan^{- 1} (3a³ tanθ - a³ tan³ θ) / (a³ - 3a³ tan² θ)

= tan^{- 1} (tan 3θ)

= 3 θ

On substituting the value of θ, we get

= 3 tan^{- 1} x/a

Therefore,

tan^{- 1} (3a² x - x³)/(a³ - 3ax²) = 3 tan^{- 1} x/a

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NCERT Solutions for Class 12 Maths - Chapter 2 Exercise 2.2 Question 10

Write the function in the simplest form : tan^{- 1} (3a² x - x³)/(a³ - 3ax²), a > 0; - a/√3 ≤ x ≤ a/√3

Summary:

The function in the simplest form : tan^{- 1} (3a² x - x³)/(a³ - 3ax²), a > 0; - a/√3 ≤ x ≤ a/√3 can be written as 3 tan^{- 1} x/a