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Equation solved with product-to-sum formulas: count or add its solutions

This type of question, asked 4 times: 2021 Paper 14, 25 Jan 2023, 7 Apr 2025, 5 Apr 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 14

    50
    The sum of all roots of sin⁡x+sin⁡2x+sin⁡3x+sin⁡4x=0\sin x+ \sin2x+ \sin3x+ \sin4x= 0 lying in [0,2π]\lbrack 0,2\pi\rbrack is

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  2. Question 2

    25 Jan 2023

    175
    If mm and nn respectively are the numbers of positive and negative values of θ\theta in the interval [−π,π]\lbrack - \pi,\pi\rbrack that satisfy the equation cos⁡2θcos⁡θ2=cos⁡3θcos⁡9θ2\cos2\theta \cos\frac{\theta}{2}= \cos3\theta \cos\frac{9\theta}{2}, then mnmn is equal to

  3. Question 3

    7 Apr 2025

    134
    The number of solutions of the equation cos⁡2θcos⁡θ2+cos⁡5θ2=2cos⁡35θ2\cos2\theta \cos\frac{\theta}{2}+ \cos\frac{5\theta}{2}= 2\cos^{3}\frac{5\theta}{2} in [−π2,π2]\left\lbrack -\frac{\pi}{2},\frac{\pi}{2} \right\rbrack is :

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  4. Question 4

    5 Apr 2026 S2

    73
    If S={θ∈[−π,π]:cos⁡θcos⁡5θ2=cos⁡7θcos⁡7θ2}S = \left\{ \theta \in \lbrack - \pi,\pi\rbrack:\cos\theta \cos\frac{5\theta}{2} = \cos7\theta\cos\frac{7\theta}{2} \right\}, then n(S)n(S) is equal to .........