Free Power Reducing Calculator

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The power-reducing identities rewrite squared trig functions — sin²θ and cos²θ — and the product sinθcosθ in terms of cos(2θ) and sin(2θ) instead, eliminating the exponent entirely. They're derived by solving the cosine double-angle formula (cos2θ = 1−2sin²θ = 2cos²θ−1) backward for sin²θ and cos²θ, so they carry exactly the same information as the double-angle identities, just rearranged for a different purpose. The main reason these formulas exist is calculus: integrating sin²x or cos²x directly has no simple antiderivative, but once rewritten as (1−cos2x)/2 or (1+cos2x)/2, the integral becomes trivial. The same substitution shows up in physics and signal processing when computing the average (RMS) power of a sinusoidal signal, since instantaneous power in an AC circuit or an oscillating wave is proportional to the square of a sine or cosine term.

How it works

Enter an angle θ. The calculator first finds cos(2θ) and sin(2θ) using the standard double-angle formulas, then substitutes them into the three power-reducing identities: sin²θ = (1 − cos2θ)/2, cos²θ = (1 + cos2θ)/2, and sinθcosθ = sin(2θ)/2. Notice the only difference between the sin² and cos² formulas is the sign in front of cos(2θ) — a detail that's easy to transpose by accident when working by hand.

Enter angle (θ).
Enter angle unit.
Click Calculate to see your results.

Examples

θ = 30°

sin²(30°) = (1 − cos60°)/2 = (1 − 0.5)/2 = 0.25, matching 0.5² = 0.25 computed directly.

θ = 45°

cos²(45°) = (1 + cos90°)/2 = (1 + 0)/2 = 0.5, matching (√2/2)² = 0.5 computed directly.

θ = 60°

sin(60°)cos(60°) = sin(120°)/2 = 0.866/2 = 0.433, matching 0.866 × 0.5 = 0.433 computed directly.

Common mistakes to avoid

Confusing the power-reducing identities with the double angle identities they're derived from — they go in opposite directions (squared term to 2θ, versus θ to 2θ).
Swapping the + and − signs between the sin²θ and cos²θ versions of the formula.
Forgetting to divide by 2 in the sinθcosθ = sin(2θ)/2 identity and using sin(2θ) directly instead.
Trying to apply the identity to sinθ or cosθ alone (without the square) — the power-reducing formulas only apply to the squared or product forms.

Frequently asked questions

sin²θ = (1 − cos2θ)/2; cos²θ = (1 + cos2θ)/2; sinθcosθ = sin(2θ)/2 — each rewrites a squared or product term using only a single, non-squared trig function of 2θ.

Most commonly in calculus, to integrate sin²x and cos²x by rewriting them in a form with no squared trig functions — a step that shows up in arc-length, area, and average-power calculations.

They run in opposite directions. The double angle identities start from θ and predict 2θ's sine/cosine; the power-reducing identities start from a squared function of θ and rewrite it in terms of 2θ — they're algebraic rearrangements of the very same relationship.

There is an analogous identity for tan²θ (in terms of cos2θ), but it's used far less often since tangent doesn't appear as frequently in the squared form that calculus integration techniques target.

No — it's an identity, meaning both sides are always exactly equal for every value of θ. The formula only changes how the value is expressed, not what the value is.

Yes — this is a standard calculus technique. Applying the identity once gives sin²θ in terms of cos2θ; squaring that result and applying the identity a second time (to the new cos²2θ term) reduces sin⁴θ down to a sum of constant, cos(2θ), and cos(4θ) terms.