Tangent zeros, poles, and period

Tangent and cotangent are already the quotients of sine and cosine. Where do those quotients meet zero, where do they blow up, and why does each graph repeat after a shift by $\pi$?

Zeros sit where the numerator does

Sine and cosine meet the axis on two lattices: sine at each integer multiple of $\pi$, cosine halfway between those multiples. The tangent is their quotient, so it meets zero where the sine does, and it blows up where the cosine does. For every complex $x$, and every integer $k$,

$$\tan x=0\text{ at }x=k\pi ,\text{a pole at }x=k\pi +\frac{\pi }{2}.$$

Every one of those zeros and poles is simple. The cotangent swaps the two lattices. For every complex $x$,

$$\cot x=0\text{ at }x=k\pi +\frac{\pi }{2},\text{a pole at }x=k\pi ,$$

and again every zero and every pole is simple.

Branches between the poles

This picture keeps $x$ real. Drag the point along $y=\tan x$. The dashed vertical lines are the poles. Between two consecutive poles the graph is one continuous branch. The blue point is the same height after a shift by $\pi$.

For cotangent the poles sit on the other lattice. Drag the point along $y=\cot x$. The dashed lines are now at the integer multiples of $\pi$. A shift by $\pi$ again keeps the height.

Period $\pi$, and odd

Away from the poles, for every complex $x$,

$$\tan (x+\pi )=\tan x,\cot (x+\pi )=\cot x.$$

That is the period $\pi$ already visible on the two pictures: the branch that starts after one pole is a copy of the branch after the previous pole.

Both functions are odd. Away from the poles, for every complex $x$,

$$\tan (-x)=-\tan x,\cot (-x)=-\cot x.$$

This picture keeps the angle real and away from the nearest poles. Drag it. The point at $-x$ sits opposite the point at $x$ through the origin.

One step rises by $1/{\cos }^2$

Away from the poles of tangent, for every complex $x$,

$$\frac{\mathrm{d}\tan x}{\mathrm{d}x}=\frac{1}{{\cos }^{2}x},$$

and away from the poles of cotangent,

$$\frac{\mathrm{d}\cot x}{\mathrm{d}x}=-\frac{1}{{\sin }^{2}x}.$$

This picture keeps $x$ real and inside one branch of the tangent, so the step never crosses a pole. Drag the point. The horizontal step has length $1$. The vertical step equals $1/{\cos }^{2}x$ at that point. The cotangent slope is the same check with a minus sign and $\sin$ in the denominator.

Tangent and cotangent meet zero on one lattice and blow up on the other, repeat every $\pi$, and are odd. Their derivatives are the squared reciprocals of cosine and sine. The next natural questions are how two angles add inside a tangent, and how a double angle rewrites.