## Definitions

### General

The inverse cosecant function, in modern notation written as arccsc(x), gives the angle θ, so that:

\[\csc \theta = x \]

Due to the periodical nature of the cosecant function, there are many angles θ that can give the same cosecant value (i.e. θ+2π, θ+4π, etc.). As a result, it is impossible to define a single inverse function, unless the range of the return values is restricted, so that a one-to-one relationship between θ and cscθ can be established. Therefore, multiple branches of the arccsc function can be defined. Commonly, the desired range of θ values spans between -π/2 and π/2. The branch of arccsc, in that case, is called the principal branch.

### Series

The arccsc function can be defined in a Taylor series form, like this:

\[ \begin{split} \textrm{arccsc}\, x & = \sum_{n=0}^{\infty}\frac{\binom{2n}{n}x^{-\left(2n+1\right)}}{4^n \left(2n+1\right)} = \\ & = \frac{1}{x} + \frac{1}{6x^3} + \frac{3}{40x^5} + \frac{5}{112x^7} \cdots \end{split} \]

The above series is valid for |x|≥1.

### Properties

The derivative of the arccsc function is:

\[ \left(\textrm{arccsc}\, x\right)' = -\frac{1}{x^2 \sqrt{1-x^{-2}}} \]

The integral of the arccsc function is given by:

\[ \int \textrm{arccsc}\, x\, \mathrm{d}x = x\, \textrm{arccsc}\, x + \ln\left(x+x\sqrt{1-x^{-2}}\right) + C \]

The following properties are also valid for the arccsc function:

\[ \begin{split} & \csc (\textrm{arccsc}\, x) &= x \\ \\ & \sin (\textrm{arccsc}\, x) &= \frac{1}{x} \\ \\ & \cos (\textrm{arccsc}\, x) &= \frac{\sqrt{x^2-1}}{x} \\ \\ & \tan (\textrm{arccsc}\, x) &= \frac{1}{\sqrt{x^2-1}} \\ \\ & \textrm{arccsc} \left(-x\right)&= - \textrm{arccsc}\, x \\ \\ & \textrm{arccsc}\, x &= \frac{\pi}{2} - \textrm{arcsec}\, x \\ \\ & \textrm{arcsec}\, x &= \arcsin\left(\frac{1}{x}\right) \\ \\ \end{split} \]