# Secant is Reciprocal of Cosine

## 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

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.17$