Trigonometric identities are equations that hold for every angle where both sides are defined. They let you rewrite expressions into simpler or more useful forms — essential for simplifying, solving equations, integrating and modeling waves. This page collects the core identities in one reference and pairs them with a checker: choose an identity and angles θ and φ, and both sides are evaluated so you can see them agree.
How to use the trigonometric identities checker
- Choose an identity from the list; they are grouped by family.
- Enter θ and, for two-angle identities such as sum or product formulas, φ. Expressions such as
pi/5are accepted. - Pick degrees or radians.
- The tape shows the left side, the right side and their difference, which should be at the level of rounding error. The table below evaluates every identity at the same angles.
Fundamental identities
Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
Quotient: tan θ = sin θ/cos θ, cot θ = cos θ/sin θ.
Pythagorean:
Even–odd: sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ.
Cofunction: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ.
Sum, difference and multiple-angle identities
| Family | Identities |
|---|---|
| Double angle | sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; tan 2θ = 2 tan θ/(1 − tan²θ) |
| Triple angle | sin 3θ = 3 sin θ − 4 sin³θ; cos 3θ = 4 cos³θ − 3 cos θ |
| Half angle | sin²(θ/2) = (1 − cos θ)/2; cos²(θ/2) = (1 + cos θ)/2; tan(θ/2) = sin θ/(1 + cos θ) = (1 − cos θ)/sin θ |
| Power reducing | sin²θ = (1 − cos 2θ)/2; cos²θ = (1 + cos 2θ)/2; tan²θ = (1 − cos 2θ)/(1 + cos 2θ) |
| Product to sum | sin θ cos φ = ½[sin(θ + φ) + sin(θ − φ)]; cos θ cos φ = ½[cos(θ − φ) + cos(θ + φ)]; sin θ sin φ = ½[cos(θ − φ) − cos(θ + φ)] |
| Sum to product | sin θ + sin φ = 2 sin((θ + φ)/2) cos((θ − φ)/2); cos θ + cos φ = 2 cos((θ + φ)/2) cos((θ − φ)/2) |
Worked example: checking sin 2θ = 2 sin θ cos θ at θ = 40°
Left side: sin 80° ≈ 0.9848077530.
Right side: 2 × sin 40° × cos 40° ≈ 2 × 0.6427876097 × 0.7660444431 ≈ 0.9848077530.
Difference: about 10−16, pure floating-point rounding. The identity holds.
Now try a false “identity” such as sin 2θ = 2 sin θ: at 40° the left side is 0.985 and the right is 1.286, so it fails immediately. A single counterexample is enough to disprove a claimed identity, which is what makes the checker useful while you work.
How to prove an identity
- Work on one side only, usually the more complicated one, and transform it until it matches the other side.
- Rewrite everything in sines and cosines when you are stuck.
- Look for a Pythagorean substitution wherever you see sin²θ, cos²θ or 1.
- Combine fractions over a common denominator, and factor differences of squares such as 1 − sin²θ.
- Never cross-multiply or move terms across as if solving an equation — that assumes what you are trying to prove.
Where identities are used
Double-angle and power-reducing formulas turn sin²x into something integrable in calculus. Sum and product formulas explain beats in acoustics, when two close frequencies add up to a pulsing tone. In navigation and surveying, the sum formulas combine bearings, and the sinusoid forms of these identities let engineers rewrite a sin x + b cos x as a single wave.
Related tools
For the values these identities rely on, see the trigonometric ratios table and the unit circle calculator. Evaluate any function at any angle with the trig functions calculator, and analyze waves built from these identities with the sinusoidal function calculator.
Frequently asked questions
What is the most important trig identity?
The Pythagorean identity sin²θ + cos²θ = 1. It comes straight from the unit circle, where (cos θ, sin θ) lies on x² + y² = 1, and the other two Pythagorean identities follow by dividing it by cos²θ or sin²θ.
Does checking an identity at one angle prove it?
No. A numerical check can disprove a false identity but cannot prove a true one, because a wrong formula can still agree at a few lucky angles. Use the checker to catch mistakes and the algebraic derivation to prove the result.
Why does the checker say undefined at some angles?
Some sides involve tangent, secant, cosecant or cotangent, which have vertical asymptotes, or a fraction whose denominator becomes zero. An identity only claims equality where both sides are defined.
How do I remember the double angle formulas?
Set φ = θ in the sum formulas. sin(θ + θ) = 2 sin θ cos θ, and cos(θ + θ) = cos²θ − sin²θ. The other two forms of cos 2θ come from replacing sin² or cos² with the Pythagorean identity.
Are the identities different in radians?
No. Identities hold for every angle in any unit. Only the numbers you type change: 90° in the cofunction identities is π/2 in radians.