Tangent is Reciprocal of Cotangent
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Theorem
Let $\theta$ be an angle such that $\sin \theta \ne 0$ and $\cos \theta \ne 0$.
Then:
- $\tan \theta = \dfrac 1 {\cot \theta}$
where $\tan$ denotes the tangent function and $\cot$ denotes the cotangent function.
Proof
\(\ds \frac 1 {\tan \theta}\) | \(=\) | \(\ds \cot \theta\) | Cotangent is Reciprocal of Tangent | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tan \theta\) | \(=\) | \(\ds \frac 1 {\cot \theta}\) |
$\tan \theta$ is not defined when $\cos \theta = 0$, and $\cot \theta$ is not defined when $\sin \theta = 0$.
$\blacksquare$