Question

Show that the range of the cotangent function is the set of all

real numbers.

Short answer

The range of cotangent function is a set of real numbers.

Step-by-step solution

Step 1. Given Information

A co-tangent function.

Step 2. Explanation

$$\begin{gathered} Consider \\ cot=\frac{x}{y} \\ a=\frac{x}{y} \\ x=ay \\ But,x^2+y^2=1 \\ {a}^2{y}^2+y^2=1 \\ y=\pm \frac{1}{\sqrt{a^2+1}} \\ thereforethevalueofxis \\ x=ay \\ =a\times \pm \frac{1}{\sqrt{a^2+1}} \\ =\pm \frac{a}{\sqrt{a^2+1}} \\ thusforsnyrealnumbera,thereisapointp(x,y)forwhichcott=a \\ i.e.-\infty <cott<\infty thusrangeofco\tan getfunctionissetofallrealnumbers. \end{gathered}$$