Let $$ x= \cos(t) - \sin(t),\quad y= 2 \sin(t). $$ I have to eliminate the parameter t. I would normally do this by solving for $\sin(t)$ and $\cos(t)$ and then use
$$\sin^2(t) +\cos^2(t) = 1,$$
but due to there being both a sine and cosine in the equation for $x$ I don't know how to solve this. Any help would be appreciated.
$$\sin(t)=\frac{y}2$$
$$\cos(t) = x+\frac{y}2$$
Hence, we have
$$\frac{y^2}4+\left( x+\frac{y}2\right)^2=1$$
$$x^2+xy+\frac{y^2}2=1$$
which is an ellipse.