Let $f\left(x\right)=1+x^2+x^3$  and $x=y+1$ , let $g$ be a function such that $\varphi \left(\theta \right)=g\left(\cos 2\theta \right)$ , then $g\left(y\right)$  and $\varphi \left(\theta \right)$  are respectively

$g\left(y\right)=f\left(x\right)=x^3+x^2+1$

          $={(y+1)}^3+{(y+1)}^2+1$

          $=y^3+4y^2+5y+3$

$\varphi \left(\theta \right)=g\left(\cos 2\theta \right)$

          $={\cos }^{3}2\theta +4{\cos }^{2}2\theta +5\cos 2\theta +3$

          $={\left(2{\cos }^2\theta -1\right)}^3+4{\left(2{\cos }^2\theta -1\right)}^2+5\left(2{\cos }^2\theta -1\right)+3$

          $=8{\cos }^6\theta +4{\cos }^4\theta +1$ .