# Secant of Supplementary Angle

From ProofWiki

## Contents

## Theorem

- $\sec \left({\pi - \theta}\right) = -\sec \theta$

where $\sec$ denotes secant.

That is, the secant of an angle is the negative of its supplement.

## Proof

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \sec \left({\pi - \theta}\right)\) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \frac 1 {\cos \left({\pi - \theta}\right)}\) | \(\displaystyle \) | \(\displaystyle \) | Secant is Reciprocal of Cosine | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \frac 1 {-\cos \theta}\) | \(\displaystyle \) | \(\displaystyle \) | Cosine of Supplementary Angle | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle -\sec \theta\) | \(\displaystyle \) | \(\displaystyle \) | Secant is Reciprocal of Cosine |

$\blacksquare$

## Also see

- Sine of Supplementary Angle
- Cosine of Supplementary Angle
- Tangent of Supplementary Angle
- Cotangent of Supplementary Angle
- Cosecant of Supplementary Angle

## Sources

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