Question 18 of 42intermediate🔧 ApplyShort Answer3 marks

Write the following functions in the simplest form: tan⁻¹[(cos x - sin x)/(cos x + sin x)], -π/4 < x < 3π/4

Correct Answer

π/4 - x

Exercise: EXERCISE 2.2 | Q: 4 | (Page 12)
For More Understanding

Explanation

We simplify the given expression by dividing numerator and denominator by cos x, obtaining (1 - tan x)/(1 + tan x). Using the tangent subtraction formula, this equals tan(π/4 - x). Since the domain ensures π/4 - x lies within (-π/2, π/2), the inverse tangent simplifies to π/4 - x.

Solution Steps

  1. Step 1: Divide numerator and denominator by cos x: tan⁻¹[(cos x - sin x)/(cos x + sin x)] = tan⁻¹[(1 - tan x)/(1 + tan x)].

  2. Step 2: Recognize that (1 - tan x)/(1 + tan x) = tan(π/4 - x).

  3. Step 3: For -π/4 < x < 3π/4, π/4 - x ∈ (-π/2, π/2), so tan⁻¹(tan(π/4 - x)) = π/4 - x.

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