The duplication formulae

What happens when the two angles of an addition formula are equal? Setting $\beta = \alpha$ in the addition formulae gives the duplication formulae, which express the trigonometric values of the double angle $2\alpha$.

Property — Duplication formulae

Substituting $\beta = \alpha$ into the addition formulae:

\sin(2\alpha) &= 2\sin\alpha\cos\alpha \\ \cos(2\alpha) &= \cos^2\alpha - \sin^2\alpha = 1 - 2\sin^2\alpha = 2\cos^2\alpha - 1 \\ \tan(2\alpha) &= \frac{2\tan\alpha}{1-\tan^2\alpha} \end{aligned}$$

The three forms of $\cos(2\alpha)$ are all equivalent (one passes from one to another using the fundamental identity $\sin^2\alpha+\cos^2\alpha=1$) and are chosen according to the context: the form in $\sin$ only and the one in $\cos$ only are particularly useful when we want to eliminate one of the two functions.

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Topics: Trigonometric formulae
Concepts: Addition formulae · Duplication formulae
Functions: Cosine · Sine · Tangent
Methods: Duplication formulae
Skills: Proving · Using formulae