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 $\sin \theta = 0$.

$\blacksquare$


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