Cotangent is Reciprocal of Tangent

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Theorem

Let $\theta$ be an angle such that $\cos \theta \ne 0$ and $\sin \theta \ne 0$.

Then:

$\cot \theta = \dfrac 1 {\tan \theta}$

where $\tan$ and $\cot$ mean tangent and cotangent respectively.


Proof

Let a point $P = \left({x, y}\right)$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.

Then:

\(\displaystyle \cot \theta\) \(=\) \(\displaystyle \frac x y\) Cotangent of Angle in Cartesian Plane
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {y / x}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\tan \theta}\) Tangent of Angle in Cartesian Plane


$\tan \theta$ is not defined when $\cos \theta = 0$, and $\cot \theta$ is not defined when $\sin \theta = 0$.

$\blacksquare$


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