Power Reducing Calculator - Trig Power Reduction Identities

A power reducing calculator rewrites a high power of a trig function — sin²θ, cos³θ, sin⁴θ, cos⁶θ, or tan²θ — into a sum of cosines or sines of integer multiples of the angle, with rational coefficients. For cos^n and even powers of sin, the reduced form uses cos(kθ) terms; for odd powers of sin, it uses sin(kθ) terms; for tan² the reduced form is a single rational expression. Pick the function, the integer power 2 through 6 (or 2 for tan), the angle, and the unit, and the page returns the original expression, the reduced identity, and a matching numeric value on both sides.

• • Precalculus and trig homework: Confirm a textbook answer for sin²θ, cos⁴θ, sin³θ, or any of the higher powers by reading the reduced identity and the matching numeric value side by side.
• • Calculus integrals of sin^n x and cos^n x: The reduction step is the first move in the standard u-substitution integral of sin^n x and cos^n x; the page shows the rewritten form.
• • Verifying double-angle and triple-angle work: When a problem rewrites cos(2θ) or cos(3θ) back to a power, the inverse path is the same identity, and the calculator confirms the two sides are equal.

The page is intentionally narrow: it covers the family of identities that turn a single trig function raised to a power into a sum of cos(kθ) or sin(kθ) terms, not product-to-sum conversions.

The output panel is built around a sanity check — the original value comes from the input, the reduced value comes from the rewritten expression, and the match row tells you whether they agree.

When a reduced form lands on a cosine term like cos(120°) or cos(2π/3) and the next step is to recover the original angle from a cosine value, the Arccos Calculator page returns the principal inverse cosine in degrees, radians, and pi form.

The page looks up the matching identity from a fixed table indexed by the function and the power, evaluates the original form on the input angle, evaluates the reduced form on the same angle, and compares the two numeric values.

• θ: The input angle, in the unit you choose (degrees or radians).
• n: The integer power (2 through 6 for sin and cos, 2 for tan).
• k: The multiple of θ in the reduced cosine or sine term: 2, 3, 4, 5, or 6.

For sin² and cos², the reduced form has a single cos(2θ) term and the constant 1, divided by 2. For sin³, the reduced form is (3 sin(θ) − sin(3θ)) / 4; for cos³, it is (3 cos(θ) + cos(3θ)) / 4. Even sin powers use cos(2θ), cos(4θ), cos(6θ); odd sin powers use sin(θ), sin(3θ), and sin(5θ); all cos powers use cos(kθ) terms.

For tan²(θ), the reduction is a single rational expression: (1 − cos(2θ)) / (1 + cos(2θ)). That formula divides by zero when cos(2θ) = −1, at θ = 90° + 180°k, and the page marks that case as undefined.

According to Wikipedia, the power-reduction identities rewrite sin^n(θ) and cos^n(θ) as linear combinations of cos(kθ) or sin(kθ) with rational coefficients, and tan²(θ) as (1 − cos(2θ)) / (1 + cos(2θ)).

According to Khan Academy, the half-angle identities sin²θ = (1 − cos(2θ))/2 and cos²θ = (1 + cos(2θ))/2 are the n = 2 case of the power-reduction family and are the same identities this calculator uses for the squared case.

When a reduced form lands on a sine term like sin(3θ) or sin(5θ) from an odd power of sin, and the next step is to recover the original angle from that sine value, the Arcsin Calculator page works in the reverse direction on the same kind of input.

Four ideas explain what the reduced form means and why the original and reduced values always agree.

These four ideas are the building blocks for the rest of the page: the half-angle identity handles the most common textbook case, the double-angle and triple-angle foundations explain the higher powers, the Pythagorean identity is the reason sin² + cos² = 1, and the domain note is the warning when tan² would divide by zero.

Once you have the reduced form, you can read off the same angles in any unit. cos(2θ) with θ = 60° is cos(120°); with θ = π/3 it is cos(2π/3). The reduced expression keeps the unit of the input.

The squared case of power reduction is the same Pythagorean identity the Right Triangle Calculator page uses for the hypotenuse and legs of a right triangle.

Five short steps cover the path from a problem statement to a verified reduced form.

1. 1 Pick the function: Choose sin, cos, or tan in the Function box. tan is only supported for power 2 on this page.
2. 2 Pick the power: Choose the integer power 2, 3, 4, 5, or 6 from the Power (n) box. The result panel updates as soon as either the function or the power changes.
3. 3 Type the angle and the unit: Type the angle in the Angle (θ) box, then pick degrees or radians. The reduced expression renders the multiples 2θ through 6θ in the same unit.
4. 4 Read the original and reduced expressions: Read the original (e.g. cos²(60°)) and the reduced (e.g. (1 + cos(120°)) / 2) from the results panel. Both are written in the same unit.
5. 5 Check the numeric match: The Original vs reduced row says Match when the two numeric values agree to the working tolerance and Mismatch when they do not.

Type sin with power 2, angle 30°, and degrees. The calculator shows sin²(30°) as the original, (1 − cos(60°)) / 2 as the reduced, both values 0.25, and a Match row. Swap the angle to 90° and the reduced value stays at 1. If the homework problem gives the angle in degrees but the reduced form in radians, the Angle Converter page switches units without retyping the angle.