Cosine in terms of Sine

Theorem
Let $x$ be a real number.

Then:

where $\sin$ denotes the real sine function and $\cos$ denotes the real cosine function.

Proof
Also, from Sign of Cosine:
 * If there exists integer $n$ such that $\left({2 n - \dfrac 1 2}\right) \pi < x < \left({2 n + \dfrac 1 2}\right) \pi$, $\cos x > 0$.
 * If there exists integer $n$ such that $\left({2 n + \dfrac 1 2}\right) \pi < x < \left({2 n + \dfrac 3 2}\right) \pi$, $\cos x < 0$.

Also see

 * Trigonometric Functions in terms of each other