Secant of Complement equals Cosecant

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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 $\theta \ne n \pi$.

$\blacksquare$


Also see


Sources