Cosine in terms of Cotangent

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

Then:

where $\cos$ denotes the real cosine function and $\cot$ denotes the real cotangent function.

Proof
It remains to determine the sign.

From Sign of Cosine:

From Sign of Cotangent:

This means:

Thus on $(1)$, $\cos x$ and $\cot x$ are the same sign:
 * $\cos x > 1$ and $\cot x > 1$

and so:


 * $\cos x = +\dfrac {\cot x} {\sqrt {1 + \cot^2 x} }$

On $(2)$, $\cos x$ and $\cot x$ are also the same sign:
 * $\cos x < 1$ and $\cot x < 1$

and so:


 * $\cos x = +\dfrac {\cot x} {\sqrt {1 + \cot^2 x} }$

On $(3)$, $\cos x$ and $\cot x$ are of opposite sign:
 * $\cos x < 1$ and $\cot x > 1$

and so:


 * $\cos x = -\dfrac {\cot x} {\sqrt {1 + \cot^2 x} }$

On $(4)$, $\cos x$ and $\cot x$ are also of opposite sign:
 * $\cos x > 1$ and $\cot x < 1$

and so:


 * $\cos x = -\dfrac {\cot x} {\sqrt {1 + \cot^2 x} }$

When $x = \paren {2 n + \dfrac 1 2} \pi$ and $x = \paren {2 n + \dfrac 3 2} \pi$, both $\cos x = 0$ and $\cot x = 0$

When $x$ is an integer $\sin x = 0$ and so $\cot x$ is undefined.

Also see

 * Trigonometric Functions in terms of each other