Secant of Angle plus Right Angle

Theorem

$\sec \left({x + \dfrac \pi 2}\right) = -\csc x$


Proof

\(\ds \sec \left({x + \frac \pi 2}\right)\) \(=\) \(\ds \frac 1 {\cos \left({x + \frac \pi 2}\right)}\) Secant is Reciprocal of Cosine
\(\ds \) \(=\) \(\ds \frac 1 {- \sin x}\) Cosine of Angle plus Right 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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