Editing Exsecant (section)

==Mathematical identities==

==={{anchor|arcexsec}}Inverse function===
The [[inverse function|inverse]] of the exsecant function, which might be symbolized {{math|arcexsec}},{{r|hall}} is well defined if its argument <math>y \geq 0</math> or <math>y \leq -2</math> and can be expressed in terms of other [[inverse trigonometric function]]s (using [[radian]]s for the angle):

<math display=block>
\operatorname{arcexsec}y = \arcsec(y+1) = \begin{cases}
{\arctan}\bigl(\!{\textstyle \sqrt{y^2+2y}}\,\bigr) & \text{if}\ \ y \geq 0, \\[6mu]
\text{undefined} & \text{if}\ \ {-2} < y < 0, \\[4mu]
\pi - {\arctan}\bigl(\!{\textstyle \sqrt{y^2+2y}}\,\bigr) & \text{if}\ \ y \leq {-2}; \\
\end{cases}_{\vphantom.}
</math>

the arctangent expression is well behaved for small angles.{{r|scheme inverse}}

===Calculus===

While historical uses of the exsecant did not explicitly involve [[calculus]], its [[derivative (mathematics)|derivative]] and [[antiderivative]] (for {{mvar|x}} in radians) are:{{r|mathworld}}

<math display=block>\begin{align}
\frac{\mathrm{d}}{\mathrm{d}x}\operatorname{exsec} x &= \tan x\,\sec x, \\[10mu]
\int\operatorname{exsec} x\,\mathrm{d}x &= \ln\bigl|\sec x + \tan x\bigr| - x + C,\vphantom{\int_|}
\end{align}</math>

where {{math|ln}} is the [[natural logarithm]]. See also [[Integral of the secant function]].

===Double angle identity===

The exsecant of twice an angle is:{{r|hall}}

<math display=block>\operatorname{exsec} 2\theta = \frac{2 \sin^2 \theta} {1 - 2 \sin^2 \theta}.</math>