Topics · 2 identities

Power Reduction

Lowering the power of squared sine and cosine using double-angle identities.

Power reduction formula: sin²θ = (1 − cos 2θ)/2

\sin^{2}\theta = \dfrac{1 - \cos 2\theta}{2}

sin²θ = (1 − cos 2θ)/2 trades a squared trig function for a plain cosine at double the angle. That is the whole point: a squared term cannot be integrated directly, but a cosine can, which makes this the standard first step for ∫ sin²θ dθ.

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Power reduction formula: cos²θ = (1 + cos 2θ)/2

\cos^{2}\theta = \dfrac{1 + \cos 2\theta}{2}

cos²θ = (1 + cos 2θ)/2 - the cosine counterpart of the sine power-reduction formula, differing only in the sign. Applied repeatedly it reduces any even power of sine or cosine down to first-degree terms.

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