Power-reducing identities calculator
Rewrite sin² or cos² of an angle as a first-power cosine of the double angle and get the number: sin² of 30 degrees is 0.25, because the cosine of 60 degrees is one half. Works with tan² too, in degrees or radians. Free in your browser.
- Free
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- No sign-up
- Your data stays on your device
Run — free
Worked example
- Func
- Sin
- Angle
- 30
- Identity
- (1 - cos(2θ)) / 2 deg
- Double angle
- 60
- Reduced
- 0.25
Power-reducing identities rewrite a squared sine or a squared cosine as a first-power cosine of the double angle, which is the form students meet right after the double-angle formulas and the form integrators want before they integrate a power of sine or cosine. Homework, contest drills, Fourier notebooks, and graphics code all need the same numeric value of sin squared of theta or cos squared of theta from one measure in degrees or radians, with a clear failure when the named function is not one of the supported identities. This capacity accepts a function name, a finite angle, and an optional unit, applies the matching power-reducing identity deterministically, and returns the reduced numeric value together with the identity string that produced it.
What power-reducing identities are and which functions they cover
In trigonometry, a power-reducing identity rewrites a squared circular function as an expression that contains only the first power of cosine of twice the original angle. The three textbook identities used here are sine squared of theta equals one minus cosine of two theta, all over two; cosine squared of theta equals one plus cosine of two theta, all over two; and tangent squared of theta equals one minus cosine of two theta over one plus cosine of two theta, whenever that denominator is nonzero.
Search intent clusters around power reducing identities, power reducing formula, sin squared identity, cos squared of theta, reduce sin squared x, and power reducing calculator. Learners want a number they can check against a homework key, not only a symbolic rewrite on a chalkboard.
The identities are algebraically equivalent to the more familiar Pythagorean pair sine squared plus cosine squared equals one, but they isolate a single squared function and replace it with a linear cosine of the double angle, which is exactly what a later integration or a Fourier coefficient needs. Tangent squared is undefined when one plus cosine of two theta is zero, which happens when theta is an odd multiple of pi over two—classic examples include ninety degrees and two hundred seventy degrees.
What you can do with it
Homework power-reducing drills
Students and graders confirm sin²(θ) and cos²(θ) for degree or radian measures, including a clear rejection when the named function is unsupported.
Integral and Fourier prep
Notebooks that rewrite a squared sine or cosine before integrating or extracting a coefficient call a deterministic power-reducing oracle instead of reimplementing the identities.
FAQ
Which power-reducing identities does this calculator apply?
sin²(θ) = (1 − cos(2θ)) / 2, cos²(θ) = (1 + cos(2θ)) / 2, and tan²(θ) = (1 − cos(2θ)) / (1 + cos(2θ)) when the denominator is nonzero. Send func as sin, cos, or tan plus a finite angle.
What happens if I send an unsupported function?
Only sin, cos, and tan are supported; names such as sec, csc, cot, or sinh are rejected and are not billed.
Does the angle have to be in degrees?
Degrees are the default. Set unit to rad when the measure is already in radians. The reduced value is unitless; unit only scales how θ is interpreted.
When is tangent squared undefined?
When 1 + cos(2θ) is zero, which happens at odd multiples of π/2 such as 90°.
Why rewrite sin squared as a cosine of the double angle?
Because the reduced form contains no squares, only the first power of the cosine of twice the angle. That is exactly the form you need before integrating a squared sine or cosine, or when working out a Fourier coefficient.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/math/power-reducing \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"func":"sin","angle":30}'const res = await fetch("https://api.kit.forhosting.com/math/power-reducing", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"func": "sin",
"angle": 30
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/power-reducing",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"func": "sin",
"angle": 30
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/power-reducing", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"func":"sin","angle":30}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"func":"sin","angle":30}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/power-reducing", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"func": "sin",
"angle": 30
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.power_reducing",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Unsupported names such as secant or cosecant fail with invalid_input and are not charged.
What power-reducing identities are and which functions they cover
Reporting that domain error, and reporting an unsupported function name such as secant or hyperbolic sine, as invalid_input is more useful than returning infinity or a silent empty payload, because answer keys and automated graders need a stable failure code rather than a non-finite JSON number that may break a downstream consumer.
How this calculator validates input, applies the identities, and shapes the response
Unit defaults to deg; set unit to rad when the measure is already in radians. Internally the angle is converted to radians only for the cosine of the double angle, then the matching identity is applied exactly as written in a textbook: sine squared uses one minus cosine of two theta over two, cosine squared uses one plus cosine of two theta over two, and tangent squared uses their ratio after a strict near-zero check on the denominator. A typical success path is func sin and angle thirty degrees, which yields reduced one quarter because cosine of sixty degrees is one half. POST a JSON body with required fields func and angle, plus an optional unit. Func must name a supported function: sin, cos, or tan. Common aliases such as sine, cosine, tangent, sin two, and sin caret two are accepted so classroom templates do not need a rewrite; any other string, including sec, csc, cot, sinh, or an empty value, fails with invalid_input that says the function is unsupported. Angle must be a finite number; JSON numbers and numeric strings are accepted; words, empty strings, null, infinity, and NaN fail before any identity runs. Results are rounded to a stable twelve-decimal precision so the same input always yields the same JSON, including for radian measures that cannot be represented exactly in binary floating point. The response echoes func, angle, and unit, reports double_angle as twice the input in the same unit, and includes identity and formula strings so a notebook can display the rewrite next to the numeric value without re-deriving which identity family was used.
Where a power-reducing API earns its keep, and what it costs
Use the free browser tool when you want an immediate reduced value while checking homework, verifying a unit-circle sketch, or confirming an integrand rewrite without installing a computer-algebra system. Results are fully deterministic: the same function, angle, and unit always yield the same reduced number. There is no language model, no external algebra service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. If you need a double-angle triple or a half-angle radical, those are different search intents and live on their own landings. Each successful request is one billable item at $0.002; failed validation—including an unsupported function name and an undefined tangent squared—does not consume balance. Accepted input is any finite real measure in degrees or radians together with one of the three supported function names; the response returns reduced as the primary numeric value, plus identity, formula, and double_angle. Limits are intentional: this endpoint does not expand higher even powers such as sine to the fourth, does not solve inverse-trig equations, and does not return exact symbolic radicals. Wire the endpoint into power-reducing drills, integral-prep worksheets, Fourier homework checkers, and animation helpers that need a squared sine or cosine as a first-power cosine of the doubled heading.
Integration use cases
Deterministic fixtures for trigonometry tests
Pipelines that seed unit tests or interactive unit-circle demos recover a stable reduced value as structured JSON without a computer-algebra dependency.
Technical questions
What does the power-reducing calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.