Cosecant of Angle plus Straight Angle

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Theorem

$\map \csc {x + \pi} = -\csc x$


Proof

\(\displaystyle \map \csc {x + \pi}\) \(=\) \(\displaystyle \frac 1 {\map \sin {x + \pi} }\) Cosecant is Reciprocal of Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {-\sin x}\) Sine of Angle plus Straight Angle
\(\displaystyle \) \(=\) \(\displaystyle -\csc x\) Cosecant is Reciprocal of Sine

$\blacksquare$


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Sources