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

• Sine of Angle plus Right Angle
• Cosine of Angle plus Right Angle
• Tangent of Angle plus Right Angle
• Cotangent of Angle plus Right Angle
• Cosecant of Angle plus Right Angle

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