# Secant of Supplementary Angle

## 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

$\blacksquare$