doubleangle

Double angle refers to trigonometric functions evaluated at twice an angle. The most common double-angle identities arise from the sum formulas for sine and cosine, together with the Pythagorean identity sin^2 x + cos^2 x = 1. The basic forms are: sin(2x) = 2 sin x cos x; cos(2x) = cos^2 x − sin^2 x, which can also be written as cos(2x) = 1 − 2 sin^2 x or cos(2x) = 2 cos^2 x − 1; and tan(2x) = 2 tan x / (1 − tan^2 x), where defined. These follow from sin(a+b) and cos(a+b) with a = b = x.

These identities relate to power-reduction formulas: sin^2 x = (1 − cos 2x)/2 and cos^2 x = (1 + cos 2x)/2, which are useful for simplifying expressions and evaluating integrals involving trig powers.

Applications of the double-angle identities include solving trigonometric equations, transforming products into sums or vice versa, and simplifying expressions in analysis and signal processing. They also underpin methods in Fourier analysis and in modeling periodic phenomena where a frequency component is doubled.

In practice, choosing among the forms for cos(2x) depends on what quantities are known or convenient. If sin x or cos x is known, the corresponding form can be advantageous; if tan x is convenient, the tan version may be preferred. Domain considerations apply for tan(2x) when the denominator 1 − tan^2 x equals zero.