If $x=\cos t,y=\cos(2t+\pi/3)$ find an analytical relation between $x$ and $y$. I'm having a bit of trouble figuring this out. At the moment this is the near solution I have:
$$y=\frac12(2\cos^2 t-1)-\sqrt{3}\sin t\cos t.$$
I should be just about to solve it but find myself stuck. I appreciate any hint on how to "eliminate" that sine, or any other way in which I could express $x$ as an expression that depends on $y$. Thanks!

You're almost there. You have correctly used that
$$
\cos\left(2t+\frac{\pi}{3}\right)=\frac{1}{2}\cos2t-\frac{\sqrt{3}}{2}\sin2t
$$
so
$$
2y=2\cos^2t-1-2\sqrt{3}\sin t\cos t=2x^2-1-2x\sqrt{3}\sin t.
$$
Thus
$$
2x\sqrt{3}\sin t=2x^2-1-2y.
$$
Hence
$$
12x^2(1-x^2)=(2x^2-1-2y)^2.
$$
