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 $\sin \theta \ne 0$.

$\ds \map \sec {\frac \pi 2 - \theta}$

$=$

$\ds \frac 1 {\map \cos {\frac \pi 2 - \theta} }$

$\ds $

$=$

$\ds \frac 1 {\sin \theta}$

$\ds $

$=$

$\ds \csc \theta$

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 $\theta \ne n \pi$.

$\blacksquare$

$\map \cos {90 \degrees - 3 \theta} = \csc 3 \theta$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I