Double-angle and reduction formulas

Taking β = α in the addition formulas gives the double-angle formulas. sin 2α = 2 sin α cos α. cos 2α = cos²α − sin²α; using the fundamental identity it can also be written cos 2α = 2cos²α − 1 or cos 2α = 1 − 2sin²α. tan 2α = 2 tan α : (1 − tan²α), where tan α and tan 2α exist. These formulas find the functions of a doubled angle when those of the angle are known, for example from sin 30° = 2 sin 15° cos 15° we get 2 sin 15° cos 15° = 1/2. The identities 1 + cos 2α = 2cos²α and 1 − cos 2α = 2sin²α are also useful for simplifying expressions. When finding sine or cosine, use the quadrant to choose the sign before the root.