Power Reduction Identities
These are extremely useful in calculus. They are just rearrangements of the double angle identities.
Cosine
\[\cos(2 \theta) = 2 \cos^2 \theta - 1 \quad\implies\quad \cos^2 \theta = \frac{1}{2}(1 + \cos(2 \theta))\]Sine
\[\cos(2 \theta) = 1 - 2 \sin^2 \theta \quad\implies\quad \sin^2 \theta = \frac{1}{2}(1 - \cos(2 \theta))\]Tangent
\[\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{1 - \cos(2 \theta)}{1 + \cos(2 \theta)}\]Secant, Cosecant, and Cotangent
There are no power reduction formulas for these functions other than just taking the reciprocal of the above identities.