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power-reducing identities — Definition, Formula & Examples

Power-reducing identities are formulas that convert squared trigonometric functions (sin²θ, cos²θ, tan²θ) into expressions involving the cosine of double the angle. They eliminate the exponent, making integration and simplification much easier.

The power-reducing identities are derived by solving the double-angle identity cos⁡2θ=1−2sin⁡2θ\cos 2\theta = 1 - 2\sin^2\theta (and its cosine counterpart) for the squared term, yielding first-degree trigonometric expressions in 2θ2\theta.

Key Formula

sin⁡2θ=1−cos⁡2θ2,cos⁡2θ=1+cos⁡2θ2,tan⁡2θ=1−cos⁡2θ1+cos⁡2θ\sin^2\theta = \frac{1 - \cos 2\theta}{2}, \quad \cos^2\theta = \frac{1 + \cos 2\theta}{2}, \quad \tan^2\theta = \frac{1 - \cos 2\theta}{1 + \cos 2\theta}
Where:
  • θ\theta = Any angle in radians or degrees
  • 2θ2\theta = Double the original angle

How It Works

Start with a squared trig function you want to simplify. Replace it using the matching power-reducing formula — this trades the square for a cosine of twice the angle. If higher even powers appear (like sin⁡4θ\sin^4\theta), apply the identity more than once to keep reducing. The result contains no squared trig functions, only first-power cosines.

Worked Example

Problem: Rewrite sin⁴θ in terms of first-power cosines (no exponents on trig functions).
Step 1: Write sin⁴θ as (sin²θ)² and apply the power-reducing identity for sin²θ.
sin⁡4θ=(1−cos⁡2θ2)2=1−2cos⁡2θ+cos⁡22θ4\sin^4\theta = \left(\frac{1 - \cos 2\theta}{2}\right)^2 = \frac{1 - 2\cos 2\theta + \cos^2 2\theta}{4}
Step 2: The cos²2θ term still has a square. Apply the power-reducing identity again, using angle 2θ.
cos⁡22θ=1+cos⁡4θ2\cos^2 2\theta = \frac{1 + \cos 4\theta}{2}
Step 3: Substitute back and simplify.
sin⁡4θ=1−2cos⁡2θ+1+cos⁡4θ24=3−4cos⁡2θ+cos⁡4θ8\sin^4\theta = \frac{1 - 2\cos 2\theta + \frac{1 + \cos 4\theta}{2}}{4} = \frac{3 - 4\cos 2\theta + \cos 4\theta}{8}
Answer: sin⁡4θ=3−4cos⁡2θ+cos⁡4θ8\sin^4\theta = \dfrac{3 - 4\cos 2\theta + \cos 4\theta}{8}

Why It Matters

In calculus, integrating sin⁡2x\sin^2 x or cos⁡2x\cos^2 x directly is not straightforward, but power-reducing identities convert them into simple cosine terms you can integrate immediately. They also appear in physics when analyzing average power in AC circuits, where squared sine and cosine functions describe instantaneous power.

Common Mistakes

Mistake: Mixing up the sign in the numerator: using 1+cos⁡2θ1 + \cos 2\theta for sin² instead of 1−cos⁡2θ1 - \cos 2\theta.
Correction: Remember: sin²θ uses the minus sign (1−cos⁡2θ1 - \cos 2\theta), and cos²θ uses the plus sign (1+cos⁡2θ1 + \cos 2\theta). A mnemonic: 'sine is sad (minus), cosine is glad (plus).'

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