Sine in terms of Tangent

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

Then:

where $\sin$ denotes the real sine function and $\tan$ denotes the real tangent function.

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

For the second part, if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$:

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

Also see

 * Trigonometric Functions in terms of each other