Secant is Reciprocal of Cosine

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Theorem

Let $\theta$ be an angle such that $\cos \theta \ne 0$.

Then:

$\sec \theta = \dfrac 1 {\cos \theta}$

where $\sec$ and $\cos$ mean secant and cosine respectively.


Proof

Let a point $P = \left({x, y}\right)$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.

Then:

\(\displaystyle \sec \theta\) \(=\) \(\displaystyle \frac r x\) $\quad$ Secant of Angle in Cartesian Plane $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {x / r}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\cos \theta}\) $\quad$ Cosine of Angle in Cartesian Plane $\quad$


When $\cos \theta = 0$, $\dfrac 1 {\cos \theta}$ is not defined.

$\blacksquare$


Sources