# Cosecant of Complement equals Secant

## Theorem

$\map \csc {\dfrac \pi 2 - \theta} = \sec \theta$ for $\theta \ne \paren {2 n + 1} \dfrac \pi 2$

where $\csc$ and $\sec$ are cosecant and secant respectively.

That is, the secant of an angle is the cosecant of its complement.

This relation is defined wherever $\cos \theta \ne 0$.

## Proof

 $\displaystyle \map \csc {\frac \pi 2 - \theta}$ $=$ $\displaystyle \frac 1 {\map \sin {\frac \pi 2 - \theta} }$ Cosecant is Reciprocal of Sine $\displaystyle$ $=$ $\displaystyle \frac 1 {\cos \theta}$ Sine of Complement equals Cosine $\displaystyle$ $=$ $\displaystyle \sec \theta$ Secant is Reciprocal of Cosine

The above is valid only where $\cos \theta \ne 0$, as otherwise $\dfrac 1 {\cos \theta}$ is undefined.

From Cosine of Half-Integer Multiple of Pi it follows that this happens when $\theta \ne \paren {2 n + 1} \dfrac \pi 2$.

$\blacksquare$