Pythagorean Identity
The Pythagorean identity is the fundamental trigonometric equation $\sin^2\theta + \cos^2\theta = 1$, with two related forms involving tangent and cotangent.
Formula
\sin^2(\theta) + \cos^2(\theta) = 1
Definition
The Pythagorean identity says that for any angle, $\sin^2(\theta) + \cos^2(\theta) = 1$. Two related forms come from dividing through by $\cos^2$ or $\sin^2$: $1 + \tan^2(\theta) = \sec^2(\theta)$ and $1 + \cot^2(\theta) = \csc^2(\theta)$. The identity is equivalent to the unit-circle definition of sin and cos and to $|e^{ix}| = 1$, the unit modulus of the complex exponential; in differential geometry, it means the curve $(\cos(t), \sin(t))$ has unit speed, since $|(-\sin(t), \cos(t))| = 1$.
Example
For $\theta = 45^\circ$: $(\sqrt{2}/2)^2 + (\sqrt{2}/2)^2 = 1/2 + 1/2 = 1$. Simplifying $\sin^4(x) - \cos^4(x) = (\sin^2(x) + \cos^2(x))(\sin^2(x) - \cos^2(x)) = \sin^2(x) - \cos^2(x)$ uses the identity together with difference of squares. To integrate $\sqrt{1 - x^2}$, substitute $x = \sin(\theta)$: then $\sqrt{1 - \sin^2(\theta)} = \cos(\theta)$, eliminating the square root.
Key Insight
This identity comes directly from the unit circle: every point $(\cos, \sin)$ is distance $1$ from the origin, so $\cos^2 + \sin^2 = 1$ by the distance formula, and knowing any one trig value lets you find the rest (with sign determined by quadrant), for example if $\sin(\theta) = 3/5$, then $\cos = \pm 4/5$. The identity is the defining relation of the unit circle as a variety, the zero set of $x^2 + y^2 - 1$; over the complex numbers this variety is a smooth rational curve isomorphic to $P^1(\mathbb{C})$.