Cosecant of Complement equals Secant

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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

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


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

$\blacksquare$


Also see


Sources