How do I eliminate the parameter (t) from x(t)=sint and y(t)=cos(2t) and rewrite it as y=f(x)

I think I can use the trig identity cos2θ=cos^2(θ)-sin^2(θ) but I'm not sure how to manipulate it.

You are on the right track. Use:
cos(2θ) = 1 - 2sin^2θ.

This follows from your identity and the fact that:
sin^2θ + cos^2θ = 1 ==> cos^2θ = 1 - sin^2θ.

So, with x = sin(t) and y = cos(2t), we have:
y = cos(2t) = 1 - 2sin^2(t) = 1 - 2x^2.

Note that this is only true for -1 < x < 1 since:
-1 < sin(t) < 1.

I hope this helps!

You are on the right track.

y(t)=cos^2(theta)-sin^2(theta)
x(t)=sin(theta)

y+2x^2=cos^2(theta)+sin^2(theta) = 1
so y+2x^2=1.

y = cos(2t) = 1-2sin^2 (t) = 1- 2x^2