Sine in terms of Secant

Theorem
Let $x$ be a real number such that $\cos x \ne 0$.

Then:

where $\sin$ denotes the sine function and $\sec$ denotes the secant function.

Proof
For the first part, if there exists integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$:

For the second part, if there exists integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$:

When $\cos x = 0$, $\sec x$ is undefined.

Also see

 * Trigonometric Functions in terms of each other