JEE Main 2025 · 7 April · evening · Single correct · +4 / −1

The number of solutions of the equation
cos 2θ cos θ/2 + cos 5θ/2 = 2 cos³ 5θ/2 in [ -π/2, π/2 ] is :

  1. (a) 5
  2. (b) 7
  3. (c) 6
  4. (d) 9
Answer and solution

Answer (b) 7

Solution

Maths sir · 4 steps

Break the product into a sum, and the right side becomes a triple-angle formula hiding in plain sight.

  1. Product to sum. Use 2 cos A cos B = cos(A+B) + cos(A-B): cos 2θ cos θ/2 = 1/2(cos 5θ/2 + cos 3θ/2). Doubling the whole equation gives 3 cos 5θ/2 + cos 3θ/2 = 4 cos³ 5θ/2.
  2. Spot triple angle. Move the 3 cos 5θ/2 across. The right side becomes 4 cos³ x - 3 cos x = cos 3x with x = 5θ/2.

    cos 3θ/2 = cos 15θ/2 ⇒ 15θ/2 = 2nπ ± 3θ/2

  3. Two families. The plus sign gives 6θ = 2nπ, so θ = nπ/3. The minus sign gives 9θ = 2nπ, so θ = 2nπ/9.
    FamilyValues in [-π/2, π/2]Count
    θ = nπ/3-π/3, 0, π/33
    θ = 2nπ/9-4π/9, -2π/9, 0, 2π/9, 4π/95
    Common to both01
  4. So. Removing the repeated θ = 0, there are 3 + 5 - 1 = 7 solutions, option (b).

The idea to keep. When you see 4 cos³ x with a stray cos x nearby, think cos 3x. When you count solutions from two families, always remove the values they share.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Trigonometric Equations

Asked about 50 times in JEE Main since 2020. This question is from 2025.