The Fundamental Identity: sin²x + cos²x = 1

Theorem (the fundamental identity). For every real number $x$,

Proof. By the unit-circle definition (Section 3), the point $P(\cos x,\sin x)$ is exactly

the point where the terminal side of $x$ meets the unit circle. But the unit circle is, by

definition, the set of all points $(X,Y)$ satisfying $X^2+Y^2=1$. Since $P=(\cos x,\sin x)$

lies on this circle for every real $x$ (the terminal side always meets the circle in exactly

one point), substituting $X=\cos x$, $Y=\sin x$ into $X^2+Y^2=1$ gives

which is the identity, usually written with the sine term first as $\sin^2x+\cos^2x=1$. Unlike

the Pythagorean proof for a right triangle (which only works for $0<x<\pi/2$), this argument

uses nothing but the equation of the circle itself, so it holds for every real $x$, including

negative angles and angles greater than $2\pi$. $\blacksquare$

Two derived identities. Two further identities, used just as often as the fundamental one

itself, follow immediately by dividing through by $\cos^2x$ or by $\sin^2x$.

Dividing by $\cos^2x$ (valid wherever $\cos x\neq0$, i.e. $x\neq(2n+1)\pi/2$):

usually written $1+\tan^2x=\sec^2x$.

Dividing by $\sin^2x$ (valid wherever $\sin x\neq0$, i.e. $x\neq n\pi$):

These three identities -- $\sin^2x+\cos^2x=1$, $1+\tan^2x=\sec^2x$, $1+\cot^2x=\text{cosec}^2x$

-- are the algebraic backbone used repeatedly in identity proofs throughout the rest of the …