Worked example

Why does cos 2x have three formulas and how do you choose?

Steps

1. cos 2x = cos²x − sin²x
   The symmetric form, straight from the sum formula. · Double-angle identity for cosine (form 1)
2. = 2cos²x − 1
   Replace sin²x by 1 − cos²x. · Double-angle identity for cosine (form 2), Pythagorean identity
3. = 1 − 2sin²x
   Replace cos²x by 1 − sin²x. · Double-angle identity for cosine (form 3), Pythagorean identity
4. Choosing
   Expression full of cos²? Use form 2. Full of sin²? Form 3. Both present? Form 1. Solving for a square (half-angle)? Rearrange forms 2 or 3.

Identities used

• 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.
• Pythagorean identity — Two squared trig terms of the same angle are added: the pair is exactly 1, so replace both with 1 — or, going the other way, replace a stubborn 1 with sin²θ + cos²θ to get a common denominator.

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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