Lowering the degree: linearisation formulae

Observation — Usefulness of the different forms of the cosine

The formulae $\cos(2\alpha) = 1-2\sin^2\alpha$ and $\cos(2\alpha) = 2\cos^2\alpha - 1$ have a crucial application: they allow us to lower the degree of $\sin^2$ and $\cos^2$, expressing them in terms of $\cos(2\alpha)$: $\sin^2\alpha = \frac{1 - \cos(2\alpha)}{2},\qquad \cos^2\alpha = \frac{1 + \cos(2\alpha)}{2}.$ These two formulae are also known as linearisation formulae: they turn a product (or a power) of sines and cosines into a sum, which is useful for integrating and for simplifying trigonometric identities.

Deriving them is immediate: it is enough to isolate $\sin^2\alpha$ in the first form and $\cos^2\alpha$ in the third form of the cosine duplication formulae. The advantage is that a second-degree term in $\sin$ or $\cos$ is replaced by a first-degree term in $\cos(2\alpha)$, which is much easier to manipulate.

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Topics: Trigonometric formulae
Concepts: Duplication formulae · Linearisation formulae
Functions: Cosine · Sine
Methods: Duplication formulae
Skills: Using formulae