An identity is an equation that is true for every value where both sides are defined, like $\sin^2 x + \cos^2 x = 1$ from the Pythagorean identity. A short list of identities lets you rewrite complicated expressions as simple ones. You do not need to memorize them as separate facts: they all come from the definitions and that one circle equation.
| Kind | Identities |
|---|---|
| Reciprocal | $\csc x = \dfrac{1}{\sin x}$, $\sec x = \dfrac{1}{\cos x}$, $\cot x = \dfrac{1}{\tan x}$ |
| Quotient | $\tan x = \dfrac{\sin x}{\cos x}$, $\cot x = \dfrac{\cos x}{\sin x}$ |
| Pythagorean | $\sin^2 x + \cos^2 x = 1$, $\ 1 + \tan^2 x = \sec^2 x$, $\ 1 + \cot^2 x = \csc^2 x$ |
| Even and odd | $\sin(-x) = -\sin x$, $\cos(-x) = \cos x$, $\tan(-x) = -\tan x$ |
Por qué funciona
The reciprocal and quotient identities are the definitions of the four other functions. Divide $\sin^2 x + \cos^2 x = 1$ by $\cos^2 x$: $\frac{\sin^2 x}{\cos^2 x} + 1 = \frac{1}{\cos^2 x}$, which is $\tan^2 x + 1 = \sec^2 x$. Divide by $\sin^2 x$ instead to get $1 + \cot^2 x = \csc^2 x$.
Idea clave
To simplify, rewrite everything in terms of $\sin x$ and $\cos x$, combine fractions, and watch for $\sin^2 x + \cos^2 x$ (or $1 - \cos^2 x$, $1 - \sin^2 x$) to replace.
Worked examples
Ejemplo resuelto: Quotient identity
Simplify $\sin x\cot x$.
- $\cot x = \dfrac{\cos x}{\sin x}$.
- $\sin x \cdot \dfrac{\cos x}{\sin x} = \cos x$, since the factor $\sin x$ cancels.
Respuesta: $\cos x$
Ejemplo resuelto: Pythagorean identity
Simplify $\dfrac{1 - \cos^2 x}{\sin x}$.
- $1 - \cos^2 x = \sin^2 x$.
- $\dfrac{\sin^2 x}{\sin x} = \sin x$.
Respuesta: $\sin x$
Ejemplo resuelto: Common denominator
Simplify $\sec x - \sin x\tan x$.
- In sines and cosines: $\dfrac{1}{\cos x} - \dfrac{\sin^2 x}{\cos x}$.
- Same denominator: $\dfrac{1 - \sin^2 x}{\cos x} = \dfrac{\cos^2 x}{\cos x}$.
- $= \cos x$.
Respuesta: $\cos x$
Error común
Cancelling a term instead of a factor: $\dfrac{1 - \sin x}{\sin x}$ is not $1 - 1$. You can only cancel something that multiplies the whole top and the whole bottom. Here the correct split is $\dfrac{1}{\sin x} - 1 = \csc x - 1$. Check any simplification by plugging in an angle such as $\frac{\pi}{6}$.