Power Reducing Calculator — sin² cos² tan²
Convert sin²(θ), cos²(θ) and tan²(θ) into lower-power cosine expressions using the standard power-reducing identities.
Unit
(1 − cos 2θ) / 2
- 1
Double the angle (2θ)
2 × 30° = 60° - 2
cos(2θ)
cos(60°) = 0.5This cosine value drives all three power-reducing identities. - 3
sin²(θ) = (1 − cos 2θ) ÷ 2
(1 − 0.5) ÷ 2 = 0.25000000
How does this calculator work?
Power-reducing identities express sin²(θ) = (1 − cos2θ)/2 and cos²(θ) = (1 + cos2θ)/2, rewriting squared trig functions as first-power cosines at double the angle. They are essential for integrating even powers of sine and cosine in calculus. Enter an angle to evaluate all three identities numerically.
Formula
How this is calculated
Power-reducing (or power-reduction) identities rewrite squared trigonometric functions — sin², cos² and tan² — as expressions that involve only the first power of a cosine at twice the angle. They are derived directly from the double-angle cosine identity: cos(2θ) = 1 − 2 sin²(θ) = 2 cos²(θ) − 1. Rearranging those two forms gives sin²(θ) = (1 − cos 2θ) / 2 and cos²(θ) = (1 + cos 2θ) / 2. Dividing gives the tan² form.
The primary use of these identities is in calculus integration: integrals of sin²(x) or cos²(x) cannot be evaluated directly by simple antidifferentiation, but after applying the power-reducing substitution they become straightforward. They also appear in Fourier analysis and signal processing when decomposing squared waveforms into fundamental frequencies.
Note: tan²(θ) is undefined whenever cos(2θ) = −1, i.e. when 2θ = π + 2kπ, equivalently θ = 90° + 180°k. At those angles the denominator 1 + cos(2θ) equals zero and the calculator shows "undefined".
Frequently asked questions
They convert squared trig functions into linear trig expressions. This is especially useful in calculus — ∫ sin²(x) dx cannot be solved directly, but substituting sin²(x) = (1 − cos 2x) / 2 makes it integrable. They also simplify Fourier and circuit analysis.
They come from the double-angle identity cos(2θ) = 1 − 2sin²(θ) = 2cos²(θ) − 1. Solving the first form for sin²(θ) gives (1 − cos 2θ)/2; solving the second for cos²(θ) gives (1 + cos 2θ)/2.
They are closely related. The half-angle identities sin(θ/2) = ±√[(1 − cosθ)/2] and cos(θ/2) = ±√[(1 + cosθ)/2] are essentially square roots of the power-reducing forms applied at half the angle.
Also known as
TG we-Calculate Editorial Team. (2026). Power Reducing Calculator — sin² cos² tan² [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/power-reducing-calculator
TG we-Calculate Editorial Team. "Power Reducing Calculator — sin² cos² tan²." TG we-Calculate. 2026. https://we-calculate.com/calculator/power-reducing-calculator.
TG we-Calculate Editorial Team, "Power Reducing Calculator — sin² cos² tan²," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/power-reducing-calculator
@misc{wecalculate_power_reducing_calculator, title = {Power Reducing Calculator — sin² cos² tan²}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/power-reducing-calculator}}, year = {2026}, note = {TG we-Calculate} }
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