[eeedycua] Tangent reflects cotangent

By definition:

cot x = 1/(tan x)

It's far from obvious from the definition that the shape of the tangent and cotangent functions are very similar, just reflected and shifted.  Usually taking the reciprocal of a function results in a very different shaped function, for example f(x)=x versus g(x)=1/f(x)=1/x.

The precise reflection and translation relationship between cotangent and tangent is

cot x = tan(pi/2 - x)

which is not an identity one sees very often listed among trigonometric identities.

We can set the right hand sides equal:

1/(tan x) = tan(pi/2 - x)
tan x = 1/(tan(pi/2 - x))

We can shift everything over by pi/4, resulting in a somewhat more symmetric-looking equation.

tan(pi/4 + x) = 1/(tan(pi/4 - x))
tan(pi/4 + x) = cot(pi/4 - x)

The period of tangent (and cotangent) is pi (not 2pi) so we could create other similar identities.

What other functions look similar to their reciprocal?  Of course a^x.  Any others?