Evaluating a value from two given equations If $x=\frac{4\lambda}{1+\lambda^2}$ and $y=\frac{2-2\lambda^2}{1+\lambda^2}$, where $\lambda$ is a real parameter and Z= $x^2$+$y^2$-xy, then what are the possible values of Z ?
How do I solve this problem? Are there any shortcuts or any theorems to be used?
I tried solving using a direct substitution, but could not proceed beyond step 2
$$ Z = x^2 + y^2 - xy$$ $$= (\frac{4\lambda}{1+\lambda^2})^2+ (\frac{2-2\lambda^2}{1+\lambda^2})^2 - (\frac{4\lambda}{1+\lambda^2})(\frac{2-2\lambda^2}{1+\lambda^2})$$  $$=\frac{4\lambda^4+8\lambda^3+8\lambda^2-8\lambda+4}{(1+\lambda^2)^2} $$

We can write $\lambda=\tan (t/2)$. Then $x=2\sin t$ and $y=2\cos t$,
so $x^2+y^2=4$. Also $xy=4\sin t\cos t=2\sin(2t)$. Which values can $4-2\sin 2t$ take?
