By "similar", I mean affinely compatible as described here. Let's say that two real functions ($f$ and $g$) are affinely compatible if and only if there exist two invertible affine functions ($k_1$, $k_2$) such that $\left(k_1 \mathop\circ f \mathop\circ k_2 \right) = g $ .
Geometrically, I think affine compatibility means that the graph of one function can be sent to the other by the composition of a translation and an axis-aligned rescaling.
Some functions are affinely compatible with their own square like $\sin$ (as shown below), $\exp$, and $\left( \lambda x \mathop. 2 ^x \right) $.
Are there any functions that are affinely compatible with their own square that are not affine functions, functions of the form $\left( \lambda x \mathop. k^x \right)$ for some arbitrary constant $k$, or themselves affinely compatible with $\sin$ ?
The other day I noticed that the graph of the real function $\sin^2$ looks very similar to the graph of $\sin$, just raised and with a different period.
Then, I stumbled on the following identities for the versine (a.k.a. versed sine or sagitta).
$$ \newcommand{\versin}{\operatorname{versin}}\versin\left(t\right) = 1-\cos\left(t\right) \tag{101} $$
$$ \versin\left(t\right) = 2\sin^2\left(\frac{t}{2}\right) \tag{102} $$
This means that (103) holds and therefore (104).
$$ 1 - \cos\left(t\right) = 2 \sin^2 \left(\frac{t}{2}\right) \tag{103} $$
$$ \frac{1 - \sin\left(\frac{\pi}{2} - 2\theta\right)}{2} = \sin^2\left(\theta\right) \tag{104} $$
(104) shows that there are two invertible affine functions $\ell_1 = \left(\lambda x \mathop. \frac{1-x}{2} \right)$ and $\ell_2 = \left(\lambda x \mathop. \frac{\pi}{2} - 2x\right) $ such that $(\ell_1 \mathop\circ \sin \mathop\circ \ell_2) =\sin^2 $ .
This proves that $\sin$ is affinely compatible with its square and that the graph of $\sin^2$ really can be produced by raising, shifting, and scrunching up the graph of $\sin$ .
I'm sort of stuck on how to determine whether the exponentials, sine-like functions, and affine functions actually hit all the functions that are affinely comaptible with their own square.