Trig identities

Trig identities are equations involving sine, cosine, tangent and the other trig functions that are true for every angle where both sides are defined. The most important is the Pythagorean identity, sin²θ + cos²θ = 1. Identities let you rewrite an expression in a simpler form, prove that two expressions are equal, find exact values and solve trig equations.

Trig identities list

Here are all the trig identities in one place. The sections below explain where each one comes from and how to use it.

Reciprocal and quotient identities

Three trig functions are defined as reciprocals of the main three: cosecant (csc), secant (sec) and cotangent (cot).

A quick way to remember the pairs: each pair has exactly one "co". Sine goes with cosecant, cosine with secant, and tangent with cotangent. The quotient identities write tangent and cotangent using sine and cosine:

Pythagorean identities

On the unit circle, the circle of radius 1 centered at the origin, the point at angle θ has coordinates (cos θ, sin θ). Drop a line from that point to the x-axis and you get a right triangle with legs cos θ and sin θ and a hypotenuse of 1.

The Pythagorean theorem on that triangle gives the first identity:

Divide every term by cos²θ, or by sin²θ, and you get the other two:

The first one is also used rearranged: sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ. Proving it from the unit circle and using it to find sin θ, cos θ or tan θ from one of the others is part of the high school Common Core standards.

Here x is the angle in radians and y is the value of sin²x + cos²x. Each part rises and falls on its own, but the sum is 1 at every angle, so the graph is a flat line. That is what makes it an identity rather than an equation.

Example 1: find cos θ from sin θ

sin θ = 3/5 and θ is between 90° and 180°. Find cos θ and tan θ.

So cos θ = ±4/5. In the second quadrant cosine is negative, so cos θ = −4/5 and tan θ = (3/5) ÷ (−4/5) = −3/4.

Even and odd identities

A negative angle is measured clockwise. Its point on the unit circle is the reflection of θ in the x-axis, which keeps the x-coordinate and flips the y-coordinate. So cosine is even and sine is odd:

Cofunction identities

The two acute angles of a right triangle add up to 90°, so the side opposite one is adjacent to the other. That gives:

In radians, replace 90° with π/2. For example, sin 20° = cos 70°. The "co" in cosine means "complement": see complementary angles.

Sum and difference identities

Watch the cosine sign: a plus on the left gives a minus on the right. These let you find exact values for angles like 15°, 75° and 105° from the special angles 30°, 45° and 60° on the unit circle.

Example 2: the exact value of tan 75°

Write 75° as 45° + 30°, and use tan 45° = 1 and tan 30° = 1/√3:

Multiply the top and bottom by √3, then rationalize the denominator:

That is about 3.732, which matches tan 75° on a calculator.

Double-angle identities

Put A = B = θ into the sum identities:

Cosine has three forms because sin²θ + cos²θ = 1 lets you swap one square for the other. Pick the form that leaves only the function you want. The double angle formula guide has more worked examples.

Example 3: sin 2θ from Example 1

With sin θ = 3/5 and cos θ = −4/5: sin 2θ = 2 × (3/5) × (−4/5) = −24/25.

Half-angle and power-reducing identities

Rearranging the cos 2θ forms gives identities that turn a square into a first power. They are essential for integrating sin²x and cos²x in calculus:

Replace θ with θ/2 and take square roots to get the half-angle identities:

The sign depends on which quadrant θ/2 is in.

How to prove trig identities

To verify an identity, start with one side, usually the more complicated one, and change it step by step until it matches the other side. Don't move terms across the equals sign: you are showing the two sides are the same, not solving an equation.

1. Work on the messier side.
2. Rewrite everything in terms of sine and cosine.
3. Look for a Pythagorean identity: sin²θ + cos²θ, 1 − sin²θ, sec²θ − 1 and so on.
4. Combine fractions over a common denominator, factor, or multiply by a conjugate.
5. Stop when you reach the other side exactly.

Example 4: (1 − cos²θ)/sin θ = sin θ

Example 5: tan θ + cot θ = sec θ csc θ

Write the left side in sine and cosine and add the fractions:

And 1/(sin θ cos θ) is (1/cos θ)(1/sin θ) = sec θ csc θ, the right side.

Example 6: (sin θ + cos θ)² = 1 + sin 2θ

Try one from the Trigonometry course: choose the simplified form, then press Mark.

Using identities to solve trig equations

Example 7: solve sin 2x = sin x for 0° ≤ x < 360°

Replace sin 2x with 2 sin x cos x, then factor. Never divide by sin x, or you lose the solutions where sin x = 0.

Either sin x = 0, so x = 0° or 180°, or cos x = 1/2, so x = 60° or 300°. The solutions are 0°, 60°, 180° and 300°. You can check them by graphing both sides in the Trigonometry calculator.

Common mistakes

• Writing sin²θ as sin θ². sin²θ means (sin θ)², the sine squared. sin(θ²) is something else.
• Splitting a sum inside a function. sin(A + B) is not sin A + sin B. Use the sum identity.
• Getting the cosine sign backward in cos(A ± B). Plus on the left, minus on the right.
• Dividing by a trig function when solving. Factor instead, or you lose solutions.
• Working on both sides at once when proving an identity. Change one side until it matches the other.
• Forgetting the ± after a square root, or picking the wrong sign for the quadrant.

Practice with a short timed paper

Trig identities come up in precalculus and calculus. The daily AP Precalculus paper is a short timed practice test of 8 questions, with a new one every day. In the UK, the same practice is A-level Maths: Algebra & Functions.

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Maths · TrigonometryCompound angle formula: sin, cos, tan (A ± B) with examplesThe compound angle formulae for sin(A ± B), cos(A ± B) and tan(A ± B): a proof, exact values, solving equations and the R cos(θ − α) form, with examples.