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Reduce powers of trig functions to first power expressions. Essential for integration.
Key insight: These formulas are derived from the double-angle formulas. They reduce the power by replacing sin² or cos² with first-power expressions.
= ∫ (1 - cos2x)/2 dx
= (1/2)∫ (1 - cos2x) dx
= (1/2)[x - sin2x/2] + C
= x/2 - sin2x/4 + C
= ∫ (1 + cos2x)/2 dx
= (1/2)∫ (1 + cos2x) dx
= (1/2)[x + sin2x/2] + C
= x/2 + sin2x/4 + C
Power reduction is essential for calculus, particularly for integrating trigonometric functions. You cannot integrate sin²x directly, but you can integrate (1-cos2x)/2.
Start with cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ. Solving for sin²θ or cos²θ gives the power reduction formulas.
Apply the formula repeatedly: sin⁶θ = (sin²θ)³. First reduce sin²θ, then expand and reduce again. Each application halves the power.