Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Quick answer: A trigonometric identity is an equation true for every allowed angle. The reciprocal, quotient and Pythagorean identities come from the unit circle, and the sum, difference and double-angle formulas let you find exact values such as sin 75° and rewrite expressions to prove other identities.
Standards: F-TF.C.8, F-TF.C.9 · Level: Grade 12 Precalculus
What are the basic trigonometric identities?
An identity is an equation that is true for every input where both sides are defined. The point (cos θ, sin θ) lies on the unit circle, so its coordinates satisfy x2 + y2 = 1.
| Type | 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 | sin2 θ + cos2 θ = 1; 1 + tan2 θ = sec2 θ; 1 + cot2 θ = csc2 θ |
How do you verify an identity?
- Pick the more complicated side and leave the other side alone.
- Rewrite everything in terms of sine and cosine.
- Combine fractions over a common denominator, expand or factor, and use a Pythagorean identity.
- Stop when the expression matches the other side.
Verify tan x + cot x = sec x csc x. Left side: sin x/cos x + cos x/sin x = (sin2 x + cos2 x)/(sin x cos x) = 1/(sin x cos x) = sec x csc x.
What are the sum, difference and double-angle formulas?
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
- tan(A ± B) = (tan A ± tan B) ÷ (1 ∓ tan A tan B)
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos2 θ − sin2 θ = 2cos2 θ − 1 = 1 − 2sin2 θ
sin 75° = sin(45° + 30°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4 ≈ 0.966. The sum formula turns two special angles into a new exact value.
Double angles follow from the sum formulas with A = B. If sin θ = 3/5 with θ in Quadrant I, then cos θ = 4/5, so sin 2θ = 2 · (3/5)(4/5) = 24/25 and cos 2θ = 16/25 − 9/25 = 7/25.
Assuming sin(A + B) = sin A + sin B. With A = B = 30°, the left side is sin 60° ≈ 0.866 but the right side is 1.
Test a suspicious identity with a special angle before spending time on a proof. One counterexample shows it is false.
- sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, 1 + cot² θ = csc² θ.
- To verify an identity, transform one side only, usually into sines and cosines.
- sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.
- sin 2θ = 2 sin θ cos θ; cos 2θ = cos² θ − sin² θ = 2cos² θ − 1 = 1 − 2sin² θ.
- sin(A + B) is not sin A + sin B; test with a special angle to catch false identities.
Proving and Using Trigonometric Identities: frequently asked questions
What is a trigonometric identity?
It is an equation involving trigonometric functions that is true for every angle where both sides are defined. For example sin² θ + cos² θ = 1 holds for every real θ, unlike an equation that has only certain solutions.
How do you verify a trigonometric identity?
Work on one side only. Rewrite in sines and cosines, combine fractions, expand or factor, and apply a Pythagorean identity until it matches the other side. Never move terms across the equal sign.
Where do the Pythagorean identities come from?
The point (cos θ, sin θ) is on the unit circle x² + y² = 1, so cos² θ + sin² θ = 1. Dividing that identity by cos² θ or sin² θ gives the tangent and cotangent versions.
Why is sin(A + B) not equal to sin A + sin B?
Sine is not a linear function. A counterexample is A = B = 30°: sin 60° ≈ 0.866, while sin 30° + sin 30° = 1. The correct formula is sin A cos B + cos A sin B.
Which form of cos 2θ should you use?
Use cos² θ − sin² θ when both values are known, 2cos² θ − 1 when only cosine is known, and 1 − 2sin² θ when only sine is known. All three are equal.
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