The Real Cotangent Function

Definition: Real Cotangent Function

The real cotangent function is defined as the ratio of the Real Cosine Function to the Real Sine Function.

$$\cot (x)\stackrel{\text{def}}{=}\frac{\cos (x)}{\sin (x)}$$

NOTE

The domain of the Real Cotangent Function is $\{x\in \mathbb{R}\mid x\ne k\pi ,k\in \mathbb{Z}\}$ because $\sin (k\pi )=0$ for all $k\in \mathbb{Z}$.

NOTATION

$$\cot (x)\mathrm{ctg}(x)\mathrm{cotg}(x)$$

Properties

Theorem: Image of the Real Cotangent Function

The image of the Real Cotangent Function is $(-\infty ;+\infty )$.

PROOF

TODO

Theorem: Periodicity of the Real Cotangent Function

The Real Cotangent Function has a period of $\pi$. More generally,

$$\cot (x+k\pi )=\cot (x)\forall k\in \mathbb{Z}$$

PROOF

TODO

Theorem: Continuity of the Real Cotangent Function

The Real Cotangent Function is continuous (everywhere it is defined).

PROOF

TODO

Theorem: Derivative of the Real Cotangent Function

The Real Cotangent Function is differentiable (everywhere it is defined) and its derivative is the negative reciprocal of the square of the Real Sine Function:

$$(\cot x{)}^{\prime }=-\frac{1}{{\sin }^{2}x}=-1-{\cot }^{2}x$$

PROOF

$$(\cot x{)}^{\prime }={\left(\frac{\cos x}{\sin x}\right)}^{\prime }=\frac{(\cos x{)}^{\prime }\sin x-(\sin x{)}^{\prime }\cos x}{{\sin }^{2}x}=\frac{-{\sin }^{2}x-{\cos }^{2}x}{{\sin }^{2}x}=\frac{-({\sin }^{2}x+{\cos }^{2}x)}{{\sin }^{2}x}=\frac{-1}{{\sin }^{2}x}$$