Pythagorean Identities

Intuition and core definition

The fundamental identity $\sin^2\theta+\cos^2\theta=1$ comes from the unit circle equation $x^2+y^2=1$. Dividing by $\cos^2\theta$ gives $1+\tan^2\theta=\sec^2\theta$ where cosine is nonzero; dividing by $\sin^2\theta$ gives $\csc^2\theta=1+\cot^2\theta$ where sine is nonzero.

Notation, language, and conditions

$\sin^2\theta$ means $(\sin\theta)^2$, not $\sin(\theta^2)$. Derived identities have domain restrictions inherited from division. Identities are equations true for every input where both sides are defined, unlike equations solved for selected angles.

Why this idea matters

Pythagorean identities translate the unit circle's distance equation into relationships among trigonometric functions, including their necessary domains.

A dependable method

1. Choose the identity containing the known and requested functions.
2. State quadrant information and relevant domains.
3. Isolate the squared unknown function.
4. Take square roots with a sign chosen from the quadrant.
5. Check the identity and a numerical or unit-circle representation.

Worked example

Common mistakes