Power Reducing Calculator
Enter an angle and this calculator applies the three power-reducing identities to rewrite sin²(x), cos²(x) and tan²(x) as expressions involving only the first power of cos(2x). It then extends the reduction to cubed and fourth powers using the same identities iterated once more. Every step is shown so you can follow the algebra and check your own work. Switch between degrees and radians at any time.
What are the power-reducing identities?
Power-reducing identities (also called power-reduction formulas) are trigonometric equations that replace a squared, cubed or higher-power trig function with an expression involving only first-power cosine at a multiple of the original angle. The three fundamental ones are: sin²(x) = (1 - cos(2x)) / 2, cos²(x) = (1 + cos(2x)) / 2, and tan²(x) = (1 - cos(2x)) / (1 + cos(2x)). All three follow directly from the double-angle identity cos(2x) = 1 - 2sin²(x) = 2cos²(x) - 1, rearranged to isolate the squared term. The value of these identities is that a squared sine or cosine cannot be integrated directly by inspection, while a single cosine at a doubled frequency can.
How to use this calculator
Select your preferred angle unit (degrees or radians), then type the angle into the field. The nine outputs update instantly: the squared, cubed and fourth-power reductions for sine, cosine and tangent. The "Show your work" panel below the result card traces every arithmetic step from the raw angle to each output, so you can follow along or spot-check your own calculations. The bars visual compares the magnitudes of the four most common outputs side by side. Note that tan²(x), tan³(x) and tan⁴(x) are undefined when cos(x) = 0, which occurs at 90°, 270° and their radian equivalents.
Extending the reduction to higher powers
Cubic and fourth-power reductions are derived by combining the squared identities with the original function. For the cube: sin³(x) = sin(x) * sin²(x) = sin(x) * (1 - cos(2x)) / 2. For the fourth power: sin⁴(x) = (sin²(x))² = ((1 - cos(2x)) / 2)² = (1 - 2cos(2x) + cos²(2x)) / 4. The last term, cos²(2x), can itself be reduced using the identity with angle 2x, giving the fully expanded form (3 - 4cos(2x) + cos(4x)) / 8. This calculator evaluates the numerical result rather than the symbolic expansion, which is what you need for definite integrals or substitutions at a specific angle.
Why these formulas matter in calculus
The most common application of power-reducing formulas is evaluating integrals of the form integral of sin^n(x) dx or cos^n(x) dx for even integer n. Because the antiderivative of cos(kx) is sin(kx)/k, replacing sin²(x) with (1 - cos(2x))/2 turns an otherwise difficult integral into a straightforward one. For example, the integral of sin²(x) dx becomes the integral of (1 - cos(2x))/2 dx = x/2 - sin(2x)/4 + C. The same strategy extends to sin⁴(x) and beyond by applying the reduction twice. In physics, the same substitution appears when computing time-averaged power in AC circuits, where voltage and current contain squared sine or cosine factors.
Power-reducing identities at common angles
| Angle | sin²(x) | cos²(x) | tan²(x) |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/4 = 0.25 | 3/4 = 0.75 | 1/3 ≈ 0.333 |
| 45° | 1/2 = 0.5 | 1/2 = 0.5 | 1 |
| 60° | 3/4 = 0.75 | 1/4 = 0.25 | 3 |
| 90° | 1 | 0 | undefined |
| 120° | 3/4 = 0.75 | 1/4 = 0.25 | 3 |
| 135° | 1/2 = 0.5 | 1/2 = 0.5 | 1 |
| 150° | 1/4 = 0.25 | 3/4 = 0.75 | 1/3 ≈ 0.333 |
| 180° | 0 | 1 | 0 |
Exact or rounded values for the three core squared identities at frequently used angles.
Frequently asked questions
What is the power-reducing formula for sin²(x)?
The standard identity is sin²(x) = (1 - cos(2x)) / 2. It comes directly from the double-angle formula cos(2x) = 1 - 2sin²(x): rearranging gives 2sin²(x) = 1 - cos(2x), then dividing both sides by 2 gives the result. For example, sin²(45°) = (1 - cos(90°)) / 2 = (1 - 0) / 2 = 0.5.
What is the power-reducing formula for cos²(x)?
The identity is cos²(x) = (1 + cos(2x)) / 2. It follows from the alternative form of the double-angle formula cos(2x) = 2cos²(x) - 1: rearranging gives 2cos²(x) = 1 + cos(2x), so cos²(x) = (1 + cos(2x)) / 2. For cos²(60°): (1 + cos(120°)) / 2 = (1 - 0.5) / 2 = 0.25, which matches the direct calculation (cos(60°))² = 0.5² = 0.25.
What is the power-reducing formula for tan²(x)?
The identity is tan²(x) = (1 - cos(2x)) / (1 + cos(2x)). It is derived from the ratio tan²(x) = sin²(x) / cos²(x) = [(1 - cos(2x)) / 2] / [(1 + cos(2x)) / 2], with the 1/2 factors cancelling. It is undefined whenever the denominator equals zero, which happens when cos(2x) = -1, that is, at x = 90°, 270°, and so on.
How do I reduce sin⁴(x) using power-reducing formulas?
Start by writing sin⁴(x) = (sin²(x))² = ((1 - cos(2x)) / 2)². Expanding the square gives (1 - 2cos(2x) + cos²(2x)) / 4. The remaining cos²(2x) can be further reduced using the same identity with angle 2x: cos²(2x) = (1 + cos(4x)) / 2. Substituting and simplifying gives the fully reduced form (3 - 4cos(2x) + cos(4x)) / 8. This calculator computes the numerical value; the symbolic expansion is shown here for reference.
Why are these formulas used in integration?
Even powers of sine and cosine do not have simple antiderivatives in their raw form. Replacing sin²(x) with (1 - cos(2x))/2 gives terms that integrate directly: the constant 1/2 integrates to x/2, and cos(2x)/2 integrates to sin(2x)/4. Without the substitution, you would need integration by parts and the result would require several extra steps. Power-reducing formulas are the standard first move for integrals like the integral of sin²(x) dx, cos²(x) dx, sin⁴(x) dx and similar.
Do these formulas work in radians as well as degrees?
Yes. The identities sin²(x) = (1 - cos(2x)) / 2 and the others hold for any angle, regardless of whether x is expressed in degrees or radians. In calculus the angle is almost always in radians, because the derivative of sin(x) equals cos(x) only when x is measured in radians. This calculator accepts both; choose the unit that matches your problem.