# Power Reducing Calculator

URL: https://speedycalculator.com/calculators/math/power-reducing/
Last updated: 2026-10-30

> The power reducing calculator evaluates sin²x, cos²x and tan²x using the power reducing formulas, which replace squares with first powers of cos 2x.

## Inputs

- Angle x (°, rad, π rad, grad, turn)

## Results

- sin²x = (1 − cos 2x)/2
- cos²x = (1 + cos 2x)/2
- tan²x = (1 − cos 2x)/(1 + cos 2x)
- cos 2x
- sin x
- cos x
- tan x

## Formula

```
sin²x = (1 − cos 2x)/2
cos²x = (1 + cos 2x)/2
tan²x = (1 − cos 2x)/(1 + cos 2x)
```

## Explanation

Power reducing formulas lower the exponent of a trigonometric expression. They are essential for integrating sin²x and cos²x, for computing averages of oscillating quantities (the average of sin² over a period is 1/2 – which is why RMS voltage is V/√2) and for Fourier analysis.

### How to use the Power Reducing Calculator

- **Angle x**

Results update as you type, and everything is calculated in your browser.

### How it works

Enter the angle x. The calculator computes cos 2x and applies each formula; the plain values sin x, cos x and tan x are shown for comparison.

### What each result means

- **sin²x = (1 − cos 2x)/2** = (1 - cos(2×rad([Angle x])))÷2
- **cos 2x** = cos(2×rad([Angle x]))
- **sin x** = sin(rad([Angle x]))
- **cos x** = cos(rad([Angle x]))

### Example values

How the main result changes when **Angle x** changes and everything else stays at the example values above:

Angle x (°)sin²x = (1 − cos 2x)/2150.06722.50.1464300.2537.50.3706450.5600.75

## Worked example

Inputs — Angle x: 30 °.

Results — **sin²x = (1 − cos 2x)/2 = 0.25**; cos²x = (1 + cos 2x)/2 = 0.75; tan²x = (1 − cos 2x)/(1 + cos 2x) = 0.3333333333; cos 2x = 0.5; sin x = 0.5; cos x = 0.8660254038; tan x = 0.5773502692.

## Key terms

- **Power reduction**: Rewriting powers of trig functions as first powers of multiple angles.

## FAQ

**What is a power reducing formula?**
An identity that rewrites sin²x, cos²x or tan²x without squares, using cos 2x.

**How do I integrate sin²x?**
∫sin²x dx = ∫(1 − cos 2x)/2 dx = x/2 − sin 2x/4 + C.

**Are there formulas for higher powers?**
Yes – apply the formulas repeatedly, e.g. sin⁴x = (3 − 4cos 2x + cos 4x)/8.

**Where do they come from?**
From the double angle formula cos 2x = 1 − 2sin²x = 2cos²x − 1.


## Sources

- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)
- [Wolfram MathWorld – Power-reduction formula](https://mathworld.wolfram.com/Power-ReductionFormula.html)