Graphing y = cot(x)*tan(x)

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f(x) = cot(x)*tan(x)

f{\left(x \right)} = \tan{\left(x \right)} \cot{\left(x \right)}

f = tan(x)*cot(x)

The graph of the function

The points of intersection with the X-axis coordinate

Graph of the function intersects the axis X at f = 0
so we need to solve the equation:

\tan{\left(x \right)} \cot{\left(x \right)} = 0

Solve this equation
The points of intersection with the axis X:

Analytical solution

x_{1} = 0

x_{2} = \frac{\pi}{2}

Numerical solution

x_{1} = 0

x_{2} = 1.5707963267949

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to cot(x)*tan(x).

\tan{\left(0 \right)} \cot{\left(0 \right)}

The result:

f{\left(0 \right)} = \text{NaN}

- the solutions of the equation d'not exist

Extrema of the function

In order to find the extrema, we need to solve the equation

\frac{d}{d x} f{\left(x \right)} = 0

(the derivative equals zero),
and the roots of this equation are the extrema of this function:

\frac{d}{d x} f{\left(x \right)} =

\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \left(- \cot^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} = 0

Solve this equation
Solutions are not found,
function may have no extrema

Inflection points

Let's find the inflection points, we'll need to solve the equation for this

\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0

(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:

\frac{d^{2}}{d x^{2}} f{\left(x \right)} =

2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)} - \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) = 0

Solve this equation
Solutions are not found,
maybe, the function has no inflections

Horizontal asymptotes

Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo

\lim_{x \to -\infty}\left(\tan{\left(x \right)} \cot{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle

Let's take the limit
so,
equation of the horizontal asymptote on the left:

y = \left\langle -\infty, \infty\right\rangle

\lim_{x \to \infty}\left(\tan{\left(x \right)} \cot{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle

Let's take the limit
so,
equation of the horizontal asymptote on the right:

y = \left\langle -\infty, \infty\right\rangle

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of cot(x)*tan(x), divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)

Let's take the limit
so,
inclined asymptote equation on the left:

y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)

\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)

Let's take the limit
so,
inclined asymptote equation on the right:

y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)

Even and odd functions

Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:

\tan{\left(x \right)} \cot{\left(x \right)} = \tan{\left(x \right)} \cot{\left(x \right)}

- Yes

\tan{\left(x \right)} \cot{\left(x \right)} = - \tan{\left(x \right)} \cot{\left(x \right)}

- No
so, the function
is
even

The graph