Cosecant of Supplementary Angle

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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

$\blacksquare$


Also see


Sources