Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
Power-reducing identities are trigonometric identities that express powers of sine and cosine in terms of first powers of cosine with double angles. These identities simplify the integration of even-powered trigonometric functions.
5 Must Know Facts For Your Next Test
Power-reducing identities are derived from the double-angle formulas for sine and cosine.
The power-reducing identity for $\sin^2(x)$ is $\sin^2(x) = \frac{1 - \cos(2x)}{2}$.
The power-reducing identity for $\cos^2(x)$ is $\cos^2(x) = \frac{1 + \cos(2x)}{2}$.
These identities are particularly useful in integrating even powers of sine or cosine functions.
Power-reducing identities help transform integrals into forms that can be solved using basic integration techniques.
Review Questions
Related terms
Double-Angle Formulas: Formulas that relate trigonometric functions of double angles to functions of single angles, such as $\cos(2x) = \cos^2(x) - \sin^2(x)$.
Integrals involving trigonometric functions that often require specific techniques like substitution, partial fractions, or power-reducing identities to solve.
$$Formulas that express trigonometric functions of half-angles in terms of the original angle, such as $$\sin(\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos(x)}{2}}$$.$$