Example: linearising sin⁴α

By applying the linearisation formulae twice, even a fourth power can be reduced to a simple sum of cosines.

Example — Identity for $\sin^4\alpha$

Let us prove the identity $\sin^4\alpha = \frac{3}{8} - \frac{1}{2}\cos(2\alpha) + \frac{1}{8}\cos(4\alpha).$

Idea. We apply the formula $\sin^2 = \dfrac{1-\cos(2\,\cdot\,)}{2}$ twice. First to the square: $\sin^4\alpha = (\sin^2\alpha)^2 = \left(\frac{1-\cos(2\alpha)}{2}\right)^2 = \frac{1 - 2\cos(2\alpha) + \cos^2(2\alpha)}{4}.$ Then to the term $\cos^2(2\alpha) = \dfrac{1+\cos(4\alpha)}{2}$: $= \frac{1}{4}\left(1 - 2\cos(2\alpha) + \frac{1+\cos(4\alpha)}{2}\right) = \frac{1}{4}\cdot\frac{2 - 4\cos(2\alpha) + 1 + \cos(4\alpha)}{2} = \boxed{\dfrac{3 - 4\cos(2\alpha) + \cos(4\alpha)}{8}}.$ which is exactly the required identity. ✓

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