Question 19 of 42intermediate🔧 ApplyShort Answer3 marks

Write the following functions in the simplest form: tan⁻¹[x/a2−x2\sqrt{a² - x²}], |x| < a

Correct Answer

sin⁻¹(x/a)

Exercise: EXERCISE 2.2 | Q: 5 | (Page 12)
For More Understanding

Explanation

This question requires simplifying the inverse trigonometric function using substitution. The key insight is to substitute x = a sin θ, which transforms the expression inside tan⁻¹ into a simple trigonometric ratio. Since |x| < a, the substitution is valid and θ lies in the principal range of sin⁻¹, ensuring cos θ > 0 for the square root simplification.

Solution Steps

  1. Step 1: Let x = a sin θ. Since |x| < a, we have |sin θ| < 1, so θ ∈ (-π/2, π/2) for principal value.

  2. Step 2: Calculate a2−x2\sqrt{a² - x²} = a2−a2sin2θ\sqrt{a² - a² sin² θ} = a2cos2θ\sqrt{a² cos² θ} = a cos θ (since cos θ > 0 in the range -π/2 < θ < π/2)

  3. Step 3: Substitute in the given function: tan⁻¹[x/a2−x2\sqrt{a² - x²}] = tan⁻¹[a sin θ/(a cos θ)] = tan⁻¹(tan θ)

  4. Step 4: Since θ ∈ (-π/2, π/2), tan⁻¹(tan θ) = θ = sin⁻¹(x/a)

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