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

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

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 $\theta \ne n \pi$.

$\blacksquare$