Secant of Complement equals Cosecant

Theorem

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

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

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

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

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

From Sine of Multiple of Pi it follows that this happens when $\theta \ne n \pi$.

Also see

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