Q.Integrate: $\displaystyle\int \cos x\cos 3x\,dx$

Why turn a product into a sum?

Products of trig functions are messy — they don't integrate nicely and are hard to simplify. Sums are clean: you can separate and integrate them term-by-term. The Product-to-Sum identities convert a product of sines and cosines into a sum (or difference), turning something like $\sin 3x \cos 5x$ into $\frac12[\sin 8x + \sin(-2x)]$.

The core idea in one sentence

The four identities

For any two angles $A$ and $B$:

The pattern:

• Same functions ($\cos\cos$ or $\sin\sin$) → result uses cosines.
• Different functions ($\sin\cos$ or $\cos\sin$) → result uses sines.
• The $\sin\sin$ case has a minus before $\cos(A+B)$ — the one that trips people up.

Where they come from (the derivation)

They follow directly from the sum and difference formulas:

Add the first two: $\sin(A+B) + \sin(A-B) = 2\sin A \cos B$. Divide by 2 → the first identity. Subtract them → the second. Add/subtract the cosine formulas → the other two.

Worked examples

Simplify $\sin 5x \cos 2x$. With $A = 5x$, $B = 2x$ (first identity):

Simplify $\sin 4x \sin x$. With $A = 4x$, $B = x$ (fourth identity):

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