### Cosecant
-$$\mathrm{cosec} \Theta = {1 \over \sin \Theta} \vert \sin \Theta \ne 0$$
+$$\operatorname{cosec} \theta = {1 \over \sin \theta} \> \vert \> \sin \theta \ne 0$$
-- **Domain** $= \mathbb{R} \ {n\pi : n \in \mathbb{Z}$
-- **Range** $= \mathbb{R} \ (-1, 1)$
-- **Turning points** at $\Theta = {{(2n + 1)\pi} \over 2} \vert n \in \mathbb{Z}$
-- **Asymptotes** at $\Theta = n\pi \vert n \in \math{Z}$
+- **Domain** $= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}$
+- **Range** $= \mathbb{R} \setminus (-1, 1)$
+- **Turning points** at $\theta = {{(2n + 1)\pi} \over 2} \> \vert \> n \in \mathbb{Z}$
+- **Asymptotes** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
### Secant
-$$\mathrm{sec} \Theta = {1 \over \cos \Theta} \vert \cos \Theta \ne =$$
+$$\operatorname{sec} \theta = {1 \over \cos \theta} \> \vert \> \cos \theta \ne 0$$
-- **Domain** $= \mathbbb{R} \ \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}$
-- **Range** $= \mathbb{R} \ (-1, 1)$
-- **Turning points** at $\Theta n \pi \vert n \in \mathbb{Z}$
-- **Asymptotes** at $\Theta = {{(2n + 1) \pi} \over 2} \vert n \in \mathb{Z}$
+- **Domain** $= \mathbb{R} \setminus \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}$
+- **Range** $= \mathbb{R} \setminus (-1, 1)$
+- **Turning points** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
+- **Asymptotes** at $\theta = {{(2n + 1) \pi} \over 2} \> \vert \> n \in \mathbb{Z}$
### Cotangent
-$$\mathrm{cot} \Theta = {{\cos \Theta} \over {\sin \Theta}}$$
+$$\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0$$
+
+- **Domain** $= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}$
+- **Range** $= \mathbb{R}$
+- **Asymptotes** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
+
+### Symmetry properties
+
+\begin{equation}\begin{split}
+ \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
+ \operatorname{sec} (-x) & = \operatorname{sec} x \\
+ \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
+ \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
+ \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
+ \operatorname{cot} (-x) & = - \operatorname{cot} x
+\end{split}\end{equation}
+
+### Complementary properties
+
+\begin{equation}\begin{split}
+ \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
+ \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
+ \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
+ \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
+\end{split}\end{equation}
+
+### Pythagorean identities
+
+\begin{equation}\begin{split}
+ 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
+ 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
+\end{split}\end{equation}