The Pythagorean Identity

On the unit circle the point is (cos θ, sin θ), and it is 1 unit from the centre.

So Pythagoras applies

The legs are cos θ and sin θ, and the hypotenuse is 1. So cos²θ + sin²θ = 1.

It is not a new fact

It is the Pythagorean theorem, written in the notation of the circle.

The quotient identity

tan θ = sin θ ÷ cos θ, because tangent is opposite over adjacent, which is y over x.

Pythagoras on a circle of radius 1

A point on the unit circle has coordinates (cos θ, sin θ) and lies at distance 1 from the origin, so cos²θ + sin²θ = 1. The identity is the distance formula applied to the circle's defining property.

Two more, by division

Dividing through by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives cot²θ + 1 = csc²θ. Both follow from the first in one step, so only one needs memorising.

What sin²θ means

sin²θ is (sin θ)², not sin(θ²) and not sin(sin θ). The exponent sits between the function name and the argument by convention, which is unusual notation and worth reading carefully.

Its main use is conversion

The identity lets you replace sin²θ with 1 − cos²θ, turning a mixed expression into one written in a single function. That conversion is the standard first move when simplifying or integrating.