Math
Trig Identities Calculator
A trig identity is an equation that is true for every angle where both sides are defined.
This page is a reference sheet and a checker in one: enter any angle and see each identity evaluated on both sides, so you can confirm a formula before using it in a proof or simplification.
Type a number, a fraction like 180/7, a π expression like 5π/6, or degrees-minutes-seconds like 30° 15′ 30″. Negative and large angles are fine.
Try an example
Result
Verification
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- sin²θ + cos²θ = 1
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- 1 + tan²θ = sec²θ
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- 1 + cot²θ = csc²θ
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- tan θ = sin θ / cos θ
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- Reciprocals (csc, sec, cot)
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- sin θ = cos(90° − θ)
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More details (5 more)
- sin(−θ) = −sin θ, cos(−θ) = cos θ
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- sin 2θ = 2 sin θ cos θ
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- sin θ
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- cos θ
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- tan θ
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
Pythagorean identities
Reciprocal and quotient identities
Cofunction and even-odd identities
Sum, difference, double, and half angle
| Identity | Formula |
|---|---|
| Sum | sin(α + β) = sin α cos β + cos α sin β; cos(α + β) = cos α cos β − sin α sin β |
| Difference | sin(α − β) = sin α cos β − cos α sin β; cos(α − β) = cos α cos β + sin α sin β |
| Tangent sum | tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β) |
| Double angle | sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ |
| Half angle | sin(θ/2) = ±√((1 − cos θ)/2); cos(θ/2) = ±√((1 + cos θ)/2) |
| Power reduction | sin²θ = (1 − cos 2θ)/2; cos²θ = (1 + cos 2θ)/2 |
| Product to sum | sin α cos β = [sin(α + β) + sin(α − β)]/2 |
How to use it
- Enter any angle in degrees or An angle unit where a full turn is 2π instead of 360 degrees.. A non-special angle such as 37° makes a good test because nothing simplifies by accident.
- Read each identity's left and right side values; a check mark means they agree.
- Use the reference tables above when writing proofs or simplifying expressions.
How to read the answer
- Both sides are rounded to six decimals for display but compared at full precision.
- Identities that involve an undefined function at your angle (for example tan 90°) are marked not defined here; they still hold wherever both sides exist.
- Numerical agreement at one angle supports an identity but does not prove it; a proof must work for every angle.
Common mistakes and edge cases
- Writing sin²θ as sin(θ²): sin²θ means (sin θ)².
- Cancelling incorrectly: sin(α + β) ≠ sin α + sin β.
- Forgetting domain restrictions: 1 + tan²θ = sec²θ is only meaningful where cos θ ≠ 0.
- Checking an identity only at 0° or 45°, where many false formulas happen to agree.
Worked examples
Non-special angle
37°
Verification
15 of 15 defined identities verified at 37°
Special angle
60°
Verification
15 of 15 defined identities verified at 60°
Radians
2π/3
Verification
15 of 15 defined identities verified at 120°
Negative angle
-135°
Verification
15 of 15 defined identities verified at -135°
Axis angle
90°: tan and sec undefined
Verification
10 of 10 defined identities verified at 90°
Zero
0°: csc and cot undefined
Verification
11 of 11 defined identities verified at 0°
Unreadable input
Verification
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Frequently asked questions
What are the three Pythagorean identities?+
sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ. The second and third come from dividing the first by cos²θ and sin²θ.
Can this calculator prove an identity?+
No. It checks both sides at one angle, which can disprove an identity (if they differ) but only supports it if they match. A proof needs algebra that works for every angle.
Why are some identities marked not defined?+
Because one side divides by zero at that angle. At 90°, tan θ and sec θ are undefined, so 1 + tan²θ = sec²θ cannot be evaluated there, even though it holds everywhere else.
What is a cofunction identity?+
A relation between a function and its co-function at complementary angles: sin θ = cos(90° − θ), tan θ = cot(90° − θ), sec θ = csc(90° − θ).
Which functions are even and which are odd?+
Cosine and secant are even (f(−θ) = f(θ)). Sine, tangent, cosecant, and cotangent are odd (f(−θ) = −f(θ)).
What is the difference between an identity and an equation?+
An identity is true for every allowed angle; an equation like sin θ = 1/2 is true only for specific angles. Identities are tools for rewriting; equations are problems to solve.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026