# Cosecant of Supplementary Angle

## Theorem

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

where $\csc$ denotes cosecant.

That is, the cosecant of an angle equals its supplement.

## Proof

 $\ds \csc \left({\pi - \theta}\right)$ $=$ $\ds \frac 1 {\sin \left({\pi - \theta}\right)}$ Cosecant is Reciprocal of Sine $\ds$ $=$ $\ds \frac 1 {\sin \theta}$ Sine of Supplementary Angle $\ds$ $=$ $\ds \csc \theta$ Cosecant is Reciprocal of Sine

$\blacksquare$