Hyperbolic tangent and cotangent

Hyperbolic sine and cosine are already the odd and even parts of the exponential. Tangent and cotangent were the quotients of sine and cosine. When is the quotient of sinh and cosh the object you need?

Sinh over cosh

Where $\cosh x\ne 0$,

$$\tanh x:=\frac{\sinh x}{\cosh x},$$

and where $\sinh x\ne 0$,

$$\coth x:=\frac{\cosh x}{\sinh x}.$$

Where both are defined, each is the reciprocal of the other. The same reciprocals of the two factors are $\mathrm{cosech}x:=1/\sinh x$ and $\mathrm{sech}x:=1/\cosh x$.

Between the lines $y=\pm 1$

This picture keeps $x$ real. Drag the input. The green height is $\tanh x$: it passes through the origin and stays strictly between the dashed lines $y=\pm 1$, approaching them as the input grows. The blue height is $\coth x$: it has a vertical asymptote at $0$ and approaches the same two lines from outside. The height at $-x$ is the opposite of the height at $x$, so both graphs are odd.

Substitute a pure imaginary argument

Substitute a pure imaginary input into tangent and cotangent. For every complex $x$ where both sides are defined,

$$\tanh x=-i\tan (ix),\coth x=i\cot (ix).$$

These two identities are the bridge between the quotients and tangent and cotangent.

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

Differentiate the quotients. Away from the poles of tanh, for every complex $x$,

$$\frac{\mathrm{d}\tanh x}{\mathrm{d}x}=\frac{1}{{\cosh }^{2}x},$$

and away from the poles of coth,

$$\frac{\mathrm{d}\coth x}{\mathrm{d}x}=-\frac{1}{{\sinh }^{2}x}.$$

Compare with the derivatives of tan and cot: the pattern is the same, with cosh and sinh in place of cos and sin. This picture keeps $x$ real. Drag the point on $y=\tanh x$. The horizontal step has length $1$. The vertical step equals $1/{\cosh }^{2}x$ at that point, and the diagonal is the tangent. The height of $\tanh$ one unit farther to the right is a different number.

The lattices pick up a factor of $i$

The zeros and the poles are simple. For every integer $k$,

The zeros, poles, and period of tangent and cotangent sit on the real lattices in the last two rows. The hyperbolic rows are those rows with an extra factor of $i$, matching the bridge above. The poles of tanh are the zeros of cosh, and the poles of coth are the zeros of sinh. Hyperbolic sine and cosine have period $2\pi i$ and no poles.

On the real line one stays strictly between $y=\pm 1$ and the other blows up at the origin. How two inputs add inside a tanh or a coth is the hyperbolic tangent addition.