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JEE Mains Maths · Trigonometric Equations

Product-to-Sum and Multiple Angles

Equations with several angles: turn products into sums to reach cos A = cos B or a product equal to zero, or collapse everything into one ratio of a multiple angle such as 3x or 4x.

All Trigonometric Equations notesJEE Mains Maths strategy

Why this matters

Twelve PYQs, half of them multiple choice, and two from 2026. Four turn products or sums of sines and cosines into cos A = cos B, or into factors equal to zero, and merge the families of roots; eight collapse to one ratio of a multiple angle through the triple-angle, double-angle or tangent-addition formulas. Two ideas cover the page.

Concept 1 of 2: Reduce to cos A = cos B

A product of two cosines becomes a sum by 2cos⁡Acos⁡B=cos⁡(A+B)+cos⁡(A−B)2\cos A\cos B=\cos(A+B)+\cos(A-B). When both sides produce the same term, it cancels and cos⁡A=cos⁡B\cos A=\cos B is left. That gives two families of roots, A=2nπ+BA=2n\pi+B and A=2nπ−BA=2n\pi-B. Count each family in the interval, then subtract the angles they share. A sum of sines works the same way: pair the terms into products and set each factor to 0.

Definition

  • 2cos⁡Acos⁡B=cos⁡(A+B)+cos⁡(A−B)2\cos A\cos B=\cos(A+B)+\cos(A-B).
  • cos⁡A=cos⁡B\cos A=\cos B exactly when A=2nπ±BA=2n\pi\pm B.
  • sin⁡C+sin⁡D=2sin⁡C+D2cos⁡C−D2\sin C+\sin D=2\sin\frac{C+D}{2}\cos\frac{C-D}{2}.
  • Total = first family + second family − common values.

Two families

cos⁡A=cos⁡B⇒A=2nπ+B or A=2nπ−B\cos A=\cos B\Rightarrow A=2n\pi+B\ \text{or}\ A=2n\pi-B

Worked example

How many θ∈[0,π]\theta\in[0,\pi] satisfy cos⁡6θcos⁡θ=cos⁡4θcos⁡3θ\cos6\theta\cos\theta=\cos4\theta\cos3\theta?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 5 Apr 2026 Shift 2 · Q73Moderate

Example 1 · Trigonometric Equations · Product-to-Sum and Multiple Angles

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 .........

Subtract the common angles

The two families overlap wherever both formulas give the same angle, and 00 is always one of them. Adding the two counts without removing the overlap overcounts.
Drill 3 more on reduce to cos a = cos b

Concept 2 of 2: Collapse to one multiple angle

Many equations are one identity away from a single ratio of 2x2x, 3x3x or 4x4x. sin⁡4x+cos⁡4x\sin^4x+\cos^4x is 1−12sin⁡22x1-\frac12\sin^22x; 4cos⁡3x−3cos⁡x4\cos^3x-3\cos x is cos⁡3x\cos3x; and tan⁡A+tan⁡B1−tan⁡Atan⁡B\frac{\tan A+\tan B}{1-\tan A\tan B} is tan⁡(A+B)\tan(A+B). Once one ratio of kxkx is left, solve for kxkx over kk times the interval, then divide by kk.

Definition

  • cos⁡3x=4cos⁡3x−3cos⁡x\cos3x=4\cos^3x-3\cos x, sin⁡3x=3sin⁡x−4sin⁡3x\sin3x=3\sin x-4\sin^3x.
  • cos⁡xcos⁡(60∘−x)cos⁡(60∘+x)=14cos⁡3x\cos x\cos(60^\circ-x)\cos(60^\circ+x)=\frac14\cos3x.
  • sin⁡4x+cos⁡4x=1−12sin⁡22x\sin^4x+\cos^4x=1-\frac12\sin^22x.
  • tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}; tan⁡A=tan⁡B\tan A=\tan B exactly when A=B+nπA=B+n\pi.
  • If x∈[a,b]x\in[a,b], then kx∈[ka,kb]kx\in[ka,kb].

Triple angle

cos⁡3x=4cos⁡3x−3cos⁡x,tan⁡3x=3tan⁡x−tan⁡3x1−3tan⁡2x\cos3x=4\cos^3x-3\cos x,\qquad \tan3x=\frac{3\tan x-\tan^3x}{1-3\tan^2x}

Worked example

Find the number and the sum of the solutions of 3sin⁡x−4sin⁡3x=123\sin x-4\sin^3x=\frac{1}{\sqrt2} in [0,π][0,\pi].
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 24 Jan 2026 Shift 2 · Q75Moderate

Example 2 · Trigonometric Equations · Product-to-Sum and Multiple Angles

The number of elements in the set {x∈[0∘,180∘]:tan⁡(x+100∘)=tan⁡(x+50∘)tan⁡xtan⁡(x−50∘)}\left\{ x \in \left\lbrack 0^{\circ},180^{\circ} \right\rbrack:\tan\left( x + 100^{\circ} \right) = \tan\left( x + 50^{\circ} \right)\tan x\tan\left( x - 50^{\circ} \right) \right\} is ____\_\_\_\_ .

Check the original tangents

Collapsing to tan⁡3x\tan3x or sin⁡4x\sin4x widens the domain. A root of the new equation can make an original tan⁡\tan or sec⁡\sec undefined; substitute each root back and drop those.
Drill 7 more on collapse to one multiple angle

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Reduce to cos A = cos B

    Two families

    cos⁡A=cos⁡B⇒A=2nπ+B or A=2nπ−B\cos A=\cos B\Rightarrow A=2n\pi+B\ \text{or}\ A=2n\pi-B
  • Collapse to one multiple angle

    Triple angle

    cos⁡3x=4cos⁡3x−3cos⁡x,tan⁡3x=3tan⁡x−tan⁡3x1−3tan⁡2x\cos3x=4\cos^3x-3\cos x,\qquad \tan3x=\frac{3\tan x-\tan^3x}{1-3\tan^2x}

Watch out for (2)

Test yourself on Trigonometric Equations

15 past JEE Mains questions from this chapter, timed at 36 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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