Secant of Supplementary Angle

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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

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

$\blacksquare$


Also see


Sources