Reciprocal Identity
Reciprocal identities express the cosecant, secant, and cotangent functions as reciprocals of sine, cosine, and tangent respectively.
Formula
\csc = \frac{1}{\sin}, \quad \sec = \frac{1}{\cos}, \quad \cot = \frac{1}{\tan}
Definition
Reciprocal identities pair each main trig function with its flip: $\csc(\theta) = 1/\sin(\theta)$, $\sec(\theta) = 1/\cos(\theta)$, $\cot(\theta) = 1/\tan(\theta)$. These are true by definition of the secondary trig functions, not separate equations to memorize, and the six trig functions form three reciprocal pairs reflecting the dual structure of the unit circle: for a point $(x, y)$ on the circle, the six values $(y, 1/y, x, 1/x, y/x, x/y)$ correspond to $(\sin, \csc, \cos, \sec, \tan, \cot)$.
Example
If $\sin(\theta) = 3/4$, then $\csc(\theta) = 4/3$; if $\cos(\theta) = 5/13$, then $\sec(\theta) = 13/5$. Simplifying $\sec(x)\sin(x) = (1/\cos(x))\sin(x) = \tan(x)$ uses the reciprocal identity for sec. The reciprocal pairing of trig functions also appears in the inverse discrete Fourier transform, where the synthesis formula uses complex exponentials that are reciprocals of those in the analysis formula.
Key Insight
"Reciprocal" means flip the fraction: cosecant, secant, and cotangent are the flipped versions of sine, cosine, and tangent, and their products with their partners always equal $1$. The reciprocal identities reflect the involution $\theta \to -\theta$ on the circle, since $\sin(-\theta) = -\sin(\theta)$ and $1/\sin(-\theta) = -\csc(\theta)$; the six trig functions organize into pairs under this and other symmetries, giving the dihedral group of order $8$ its natural action on the set of six trig functions.