Cosecant of Angle plus Straight Angle

Theorem

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


Proof

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

$\blacksquare$


Also see


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
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