Shape of Secant Function

Theorem
The nature of the secant function on the set of real numbers $\R$ is as follows:


 * $(1): \quad$ $\sec x$ is continuous and strictly increasing on the intervals $\left[{0 \,.\,.\, \dfrac \pi 2}\right)$ and $\left({\dfrac \pi 2 \,.\,.\, \pi}\right]$
 * $(2): \quad$ $\sec x$ is continuous and strictly decreasing on the intervals $\left[{- \pi \,.\,.\, - \dfrac \pi 2}\right)$ and $\left({- \dfrac \pi 2 \,.\,.\, 0}\right]$
 * $(3): \quad$ $\sec x \to + \infty$ as $x \to - \dfrac \pi 2^+$
 * $(4): \quad$ $\sec x \to + \infty$ as $x \to \dfrac \pi 2^-$
 * $(5): \quad$ $\sec x \to - \infty$ as $x \to \dfrac \pi 2^+$
 * $(6): \quad$ $\sec x \to - \infty$ as $x \to \dfrac {3 \pi} 2^-$

Proof
From Derivative of Secant Function: $D_x \left({\sec x}\right) = \dfrac{\sin x}{\cos^2 x}$

From Sine and Cosine are Periodic on Reals/Corollary: $\forall x \in \left({- \pi \,.\,.\, \pi}\right) \setminus \left\{- \dfrac \pi 2, \dfrac \pi 2\right\}: \cos x \ne 0$.

Thus, from Square of Element of Ordered Integral Domain is Positive: $\forall x \in \left({- \pi \,.\,.\, \pi}\right) \setminus \left\{- \dfrac \pi 2, \dfrac \pi 2\right\}: \cos^2 x > 0$.

From Sine and Cosine are Periodic on Reals/Corollary: $\sin x > 0$ on the open interval $\left({0 \,.\,.\, \pi}\right)$.

It follows that $\forall x \in \left({0 \,.\,.\, \pi}\right) \setminus \left\{\dfrac \pi 2\right\}: \dfrac{\sin x}{\cos^2 x} > 0$.

From Sine and Cosine are Periodic on Reals/Corollary: $\sin x < 0$ on the open interval $\left({-\pi \,.\,.\, 0}\right)$.

It follows that $\forall x \in \left({-\pi \,.\,.\, 0}\right) \setminus \left\{- \dfrac \pi 2\right\}: \dfrac{\sin x}{\cos^2 x} < 0$.

Thus, $(1)$ and $(2)$ follow from Derivative of Monotone Function and Differentiable Function is Continuous.

From Zeroes of Sine and Cosine: $\cos - \dfrac \pi 2 = \cos \dfrac \pi 2 = \cos \dfrac {3 \pi} 2 = 0$.

From Sine and Cosine are Periodic on Reals/Corollary, $\cos x > 0$ on the open interval $\left({- \dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

From the same source, $\cos x < 0$ on the open interval $\left({\dfrac \pi 2 \,.\,.\, \dfrac {3 \pi} 2}\right)$.

Thus, $(3)$, $(4)$, $(5)$ and $(6)$ follow from Infinite Limit Theorem.

Also see

 * Shape of Sine Function
 * Shape of Cosine Function
 * Shape of Tangent Function
 * Shape of Cotangent Function
 * Shape of Cosecant Function