Cosecant of Angle plus Three Right Angles

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Theorem

$\map \csc {x + \dfrac {3 \pi} 2} = -\sec x$


Proof

\(\ds \map \csc {x + \frac {3 \pi} 2}\) \(=\) \(\ds \frac 1 {\map \sin {x + \frac {3 \pi} 2} }\) Cosecant is Reciprocal of Sine
\(\ds \) \(=\) \(\ds \frac 1 {-\cos x}\) Sine of Angle plus Three Right Angles
\(\ds \) \(=\) \(\ds -\sec x\) Secant is Reciprocal of Cosine

$\blacksquare$


Also see


Sources