 Solution

The even-power family is dense on $[1,2]$. It is the algebra $A=\{p(t^2):p\in \mathbb{C}[s]\}$. It contains constants, is closed under complex conjugation, and separates points because the real function $t\mapsto t^2$ is injective on this positive interval. The complex Stone-Weierstrass theorem therefore applies.

Equivalently, the squaring map is a homeomorphism from $[1,2]$ onto $[1,4]$. Given $f\in C([1,2])$, approximate the continuous function $s\mapsto f(\sqrt{s})$ uniformly on $[1,4]$ by complex polynomials, using the Weierstrass approximation theorem on the real and imaginary parts. Substitution $s=t^2$ produces exactly the required even-power approximants. The positive interval is what lets even powers distinguish all its points.