Properties

Theorem

The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \sec(z)=\tan(z)\sec(z),$$ where $\sec$ denotes the secant and $\cot$ denotes the cotangent.

Proof

From the definition of secant, $$\sec(z) = \dfrac{1}{\cos(z)},$$ and so using the quotient rule, the derivative of cosine, and the definition of tangent, $$\dfrac{\mathrm{d}}{\mathrm{d}z} \sec(z) = \dfrac{\mathrm{d}}{\mathrm{d}z} \dfrac{1}{\cos(z)} = \dfrac{\sin(z)}{\cos^2(z)}=\tan(z)\sec(z),$$ as was to be shown. $\blacksquare$

References

1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.3.109$

Theorem

The following formula holds: $$\sec(\mathrm{gd}(x))=\cosh(x),$$ where $\sec$ denotes the secant, $\mathrm{gd}$ denotes the Gudermannian, and $\cosh$ denotes the hyperbolic cosine.

Proof

References

Theorem

The following formula holds: $$\cosh(\mathrm{gd}^{-1}(x))=\sec(x),$$ where $\cosh$ is the hyperbolic cosine, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\sec$ is the secant.