Question 17 of 42intermediate🔧 ApplyShort Answer3 marks

Write the following functions in the simplest form: tan⁻¹[(1+x2\sqrt{1+x²} - 1)/x], x ≠\neq 0

Correct Answer

The simplest form is ½ tan⁻¹x.

Exercise: EXERCISE 2.2 | Q: 3 | (Page 12)
For More Understanding

Explanation

This question from Exercise 2.2 requires simplifying the inverse trigonometric function using substitution. The key insight is recognizing that the expression 1+x2\sqrt{1+x²} suggests a trigonometric substitution. Following the pattern shown in Example 5 of the textbook, we substitute x = tan θ to simplify the expression. The algebraic manipulation using half-angle formulas transforms the complex fraction into a simple tangent function, which then simplifies through the inverse tangent property.

Solution Steps

  1. Step 1: Let x = tan θ. Then 1+x2\sqrt{1+x²} = 1+tan2θ\sqrt{1+tan²θ} = sec2θ\sqrt{sec²θ} = sec θ

  2. Step 2: Substitute in the given function: tan⁻¹[(sec θ - 1)/tan θ] = tan⁻¹[(1/cos θ - 1)/(sin θ/cos θ)] = tan⁻¹[(1 - cos θ)/sin θ]

  3. Step 3: Using half-angle formulas: 1 - cos θ = 2sin²(θ/2) and sin θ = 2sin(θ/2)cos(θ/2)

  4. Step 4: Therefore, tan⁻¹[(2sin²(θ/2))/(2sin(θ/2)cos(θ/2))] = tan⁻¹[sin(θ/2)/cos(θ/2)] = tan⁻¹[tan(θ/2)] = θ/2

  5. Step 5: Since x = tan θ, we have θ = tan⁻¹x. Thus, the simplest form is ½ tan⁻¹x

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