The Pythagorean identity

The trigonometry course showed where sin and cos come from. Pre-calculus puts them to work as functions you can manipulate — and that starts with the one identity everything else is built on.

It's just Pythagoras on the unit circle

Place a point P on the unit circle at angle θ. By definition, its coordinates are (cos θ, sin θ). Drop a vertical line and you've made a right triangle:

• the horizontal leg has length cos θ
• the vertical leg has length sin θ
• the hypotenuse is the radius, length 1

The Pythagorean theorem (leg² + leg² = hyp²) reads, on this triangle:

sin²θ + cos²θ = 1

That's the Pythagorean identity. It isn't a new fact to memorize — it's the oldest theorem in geometry, applied to a triangle of radius 1. (Note the shorthand: sin²θ means (sin θ)².)

The two it generates

Divide the whole identity by cos²θ, and again by sin²θ, and two more identities drop out for free:

So the single circle relation seeds all three Pythagorean identities.

What identities are for

An identity is true for every angle, which makes it a legal swap you can make anywhere. Their main job is to rewrite an expression into a form you can actually work with:

• Simplify: 1 − sin²θ is just cos²θ.
• Solve equations: replace sin²θ with 1 − cos²θ so everything is in one function.

Why this matters

The Pythagorean identity is the most-used line in all of trigonometry. In calculus it's the key that unlocks integrals full of sines and cosines — you swap sin²θ for 1 − cos²θ (or use sec²θ = tan²θ + 1) to turn an impossible-looking integral into a routine one.