Trigonometric Identity
A trigonometric identity is an equation involving trigonometric functions that is true for all valid values of the variable.
Definition
A trigonometric identity is an equation involving sine, cosine, or tangent that holds for all values of the variable for which both sides are defined. Major categories include Pythagorean identities, reciprocal identities, quotient identities, and co-function identities (such as $\sin(90^\circ - \theta) = \cos(\theta)$). They arise from the properties of the exponential function via Euler's formula, $e^{ix} = \cos(x) + i\sin(x)$; the sum formula $\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)$ follows from $e^{i(a+b)} = e^{ia}e^{ib}$, making complex multiplication the generating mechanism for all angle-addition identities.
Example
$\sin^2(\theta) + \cos^2(\theta) = 1$ is a trig identity; plugging in $30^\circ$ gives $(0.5)^2 + (\sqrt{3}/2)^2 = 1$, always true. To prove $\tan(\theta)\cos(\theta) = \sin(\theta)$: the left side $= (\sin/\cos)\cos = \sin$, so both sides match wherever $\cos(\theta) \neq 0$. The double-angle identity $\cos(2x) = \cos^2(x) - \sin^2(x)$ follows from $e^{2ix} = (\cos(x) + i\sin(x))^2$, taking the real part.
Key Insight
Identities are shortcuts: instead of doing complicated trig calculations from scratch, you can swap one expression for an equal one to simplify your work, and they are used to simplify integrals, solve trig equations, and transform expressions. Every trigonometric identity can be derived from just two facts, Euler's formula and the property $e^{a+b} = e^a e^b$, reducing the entire algebra of trig identities to complex exponential arithmetic.