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
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 $\theta \ne \paren {2 n + 1} \dfrac \pi 2$.

Also see

 * Sine of Complement equals Cosine
 * Cosine of Complement equals Sine
 * Tangent of Complement equals Cotangent
 * Cotangent of Complement equals Tangent
 * Secant of Complement equals Cosecant