Worked example

Derive the double-angle formulas from the sum formulas

Steps

1. sin(A + B), B = A
   Specialise the sine sum. · Sine of a sum
2. sin 2A = 2 sin A cos A
   Two identical terms.
3. cos(A + B), B = A
   Specialise the cosine sum. · Cosine of a sum
4. cos 2A = cos²A − sin²A
   First form. · Double-angle identity for cosine (form 1)
5. sin²A = 1 − cos²A (and the reverse)
   Pythagorean substitution gives the other two forms. · Pythagorean identity
6. cos 2A = 2cos²A − 1 = 1 − 2sin²A
   Forms 2 and 3. · Double-angle identity for cosine (form 2), Double-angle identity for cosine (form 3)

Identities used

• Sine of a sum — The angle itself is a sum and you need the functions separately — to evaluate exactly (sin 75° = sin(45° + 30°)), to expand, or to recognise the pattern running backwards.
• Cosine of a sum — Expanding a cosine of a sum — and the reason cos 2A and the power-reducing formulas look the way they do. The sign here is minus, which surprises most students the first time.
• Double-angle identity for sine — The question is in 2x and the answer (or the next step) is in x — or you need to integrate/solve something containing sin x cos x, which is this formula run backwards.
• Double-angle identity for cosine (form 1) — The symmetric form — use it when both sin² and cos² are already present and you want to collapse them into one cosine.
• Double-angle identity for cosine (form 2) — Cosine-only version: use it when the expression contains cos² and you want a single cosine, and as the source of the cos² power-reducing formula.
• Double-angle identity for cosine (form 3) — Sine-only version. It is the one to reach for when sin² is in the way, and the parent of the sin² power-reducing formula used constantly in calculus.

Check it yourself

Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.

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