|
Theorem
The following formula holds:
$$\mathrm{arctan}(z) = \mathrm{arccot}\left( \dfrac{1}{z} \right),$$
where $\mathrm{arctan}$ denotes the inverse tangent and $\mathrm{arccot}$ denotes the inverse cotangent.
Proof
Let $y = \arctan \left( \dfrac{1}{z} \right)$. Then since arctan is the inverse function of tangent,
$$\tan(y)=\dfrac{1}{z}.$$
By the definition of cotangent, we get
$$\cot(y)=z.$$
Since $\mathrm{arccot}$ is the inverse function of $\cot$, take the $\mathrm{arccot}$ of each side to get
$$y = \mathrm{arccot}(z).$$
Therefore we have shown
$$\arctan \left( \dfrac{1}{z} \right) = \mathrm{arccot}(z),$$
as was to be shown.
References
|