Trigonometric Identities

You’ll be able to

State and apply the Pythagorean identity
Rewrite tangent and the reciprocal functions in terms of sine and cosine
Simplify a trigonometric expression using basic identities

The Pythagorean identity

Because every point (cos θ, sin θ) lies on the unit circle, its coordinates satisfy x² + y² = 1. Substituting gives the master identity of trigonometry: sin²θ + cos²θ = 1. It holds for every angle θ, and from it you can find sine given cosine (or vice versa), up to sign.

Tangent and the reciprocals

The other trig functions are built from sine and cosine. Tangent is their ratio: tan θ = sin θ / cos θ. The three reciprocal functions are secant (1/cos θ), cosecant (1/sin θ), and cotangent (cos θ / sin θ, the reciprocal of tangent). Rewriting everything in terms of sin and cos is the surest way to simplify a messy expression.

Core identities

sin²θ + cos²θ = 1 · tan θ = sin θ / cos θ · cot θ = cos θ / sin θ

Dividing the Pythagorean identity by cos²θ gives tan²θ + 1 = sec²θ; dividing by sin²θ gives 1 + cot²θ = csc²θ.

Worked example

Simplify the expression sin θ / cos θ, and then simplify sin²θ + cos²θ.

1.By definition, the ratio of sine to cosine is tangent: sin θ / cos θ = tan θ.
2.The sum sin²θ + cos²θ is the Pythagorean identity.
3.That sum equals 1 for every angle θ.

Answer: sin θ / cos θ = tan θ, and sin²θ + cos²θ = 1. The first is the definition of tangent; the second is the Pythagorean identity.

Checkpoint

Which equation is the Pythagorean identity?

Watch out

Keep the squares in place: it is sin²θ + cos²θ = 1, not sin θ + cos θ = 1. The un-squared sum varies with θ (it reaches √2 at 45°), so only the squared version is an identity.

Checkpoint

Simplify sin θ / cos θ.

On the exam

When a proof stalls, convert every function to sine and cosine and look for the Pythagorean identity. Most AP simplification problems collapse once everything is written over a common sin/cos foundation.

Answer the 2 checkpoints as you read.