Power-reducing identities

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

1. Power-reducing identities are derived from the double-angle formulas for sine and cosine.
2. The power-reducing identity for $\sin^2(x)$ is $\sin^2(x) = \frac{1 - \cos(2x)}{2}$.
3. The power-reducing identity for $\cos^2(x)$ is $\cos^2(x) = \frac{1 + \cos(2x)}{2}$.
4. These identities are particularly useful in integrating even powers of sine or cosine functions.
5. Power-reducing identities help transform integrals into forms that can be solved using basic integration techniques.

Review Questions

• What is the power-reducing identity for $\sin^2(x)$?
• How can power-reducing identities simplify the integration of trigonometric functions?
• What is the importance of double-angle formulas in deriving power-reducing identities?

Related terms

Double-Angle Formulas: Formulas that relate trigonometric functions of double angles to functions of single angles, such as $\cos(2x) = \cos^2(x) - \sin^2(x)$.

Trigonometric Integrals: Integrals involving trigonometric functions that often require specific techniques like substitution, partial fractions, or power-reducing identities to solve.

Half-Angle Formulas: $$Formulas that express trigonometric functions of half-angles in terms of the original angle, such as $$\sin(\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos(x)}{2}}$$.$$