Cosine in terms of Cotangent
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Theorem
Let $x$ be a real number such that $\cos x \ne 0$.
Then:
| \(\ds \cos x\) | \(=\) | \(\ds +\frac {\cot x} {\sqrt {1 + \cot^2 x} }\) | if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$ | |||||||||||
| \(\ds \cos x\) | \(=\) | \(\ds -\frac {\cot x} {\sqrt {1 + \cot^2 x} }\) | if there exists an integer $n$ such that $\paren {2 n - 1} \pi < x < 2 n \pi$ |
where $\cos$ denotes the real cosine function and $\cot$ denotes the real cotangent function.
Proof
| \(\ds \cos x\) | \(=\) | \(\ds \pm \frac 1 {\sqrt {1 + \tan^2 x} }\) | Cosine in terms of Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \pm \frac 1 {\sqrt {1 + \frac 1 {\cot^2 x} } }\) | Cotangent is Reciprocal of Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \pm \frac {\cot x} {\sqrt {1 + \cot^2 x} }\) | multiplying denominator and numerator by $\cot x$ |
It remains to determine the sign.
From Sign of Cosine:
| \(\ds \cos x\) | \(>\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \cos x\) | \(<\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ |
From Sign of Cotangent:
| \(\ds \cot x\) | \(>\) | \(\ds 0\) | if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \cot x\) | \(<\) | \(\ds 0\) | if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ |
This means:
| \(\text {(1)}: \quad\) | \(\ds \cot x\) | \(>\) | \(\ds 0\) | when $2 n \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds \cot x\) | \(<\) | \(\ds 0\) | when $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + 1} \pi$ | ||||||||||
| \(\text {(3)}: \quad\) | \(\ds \cot x\) | \(>\) | \(\ds 0\) | when $\paren {2 n + 1} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ | ||||||||||
| \(\text {(4)}: \quad\) | \(\ds \cot x\) | \(<\) | \(\ds 0\) | when $\paren {2 n + \dfrac 3 2} \pi < x < \paren {2 n + 2} \pi$ |
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.
$\blacksquare$