Double angles

Double the angle, not the value: sin 2θ is NOT 2 sin θ. The correct double-angle formula is sin 2θ = 2 sin θ cos θ.

A very common slip: Writing sin 2θ = 2 sin θ loses the cos θ factor — always include both.

One formula, three faces: cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1. The three are equal (via sin²+cos²=1); pick whichever matches what you know.

Which form?: Know only sin? Use 1 − 2sin²θ. Know only cos? Use 2cos²θ − 1. Know both? Any form works.

Get sin θ and cos θ first: To find sin 2θ or cos 2θ, first find sin θ and cos θ (often via the Pythagorean identity), then substitute into the double-angle formula.

Pick the form that avoids extra work: Here using 2cos²θ − 1 means you never need sin θ at all — choose the form that uses what you're given.

Spot the double-angle pattern: When an expression has 2 sin θ cos θ or cos²θ − sin²θ, replace it with sin 2θ or cos 2θ. Difference-of-squares + the Pythagorean identity often reveal it.

Factor, then identify: Factor where you can; an expression like cos²θ − sin²θ is exactly cos 2θ.