We know that a monotonic transformation of a utility function still represents the same preferences and as the old utility function represented homothetic preferences the new one does, too.
As an easy example you could look at Cobb-Douglas utility functions of the form $u(x,y) = a\left(x y\right)^\alpha$. For $\alpha = \frac12$ the utility function is homogeneous of degree 1, but for every $\alpha,a>0,$ the preference relation is homogeneous of degree 1.
We never used homothetic as a property of the utility function to answer your question (as it works for every utility function).
A homothetic utility function is, to the best of my knowledge, not very clearly defined, I have see two different versions:
- Homothetic as different name for homogeneous of degree 1
- A homothetic utility function is a utility function that represents
a homothetic preference relation.
Which means, with definition 2 Cobb-Douglas utility functions with equal weights are always homothetic, with definition 2 only some of them are:
$$u(tx,ty) = (tx)^\frac12 (ty)^\frac12 = t xy = t u(x,y)$$
and some are not
$$u(tx,ty) = (tx) (ty) = t^2 xy \neq t u(x,y)$$