Calculus II

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Power-reducing identities

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Calculus II

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.

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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?

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