To simplify the expression $cos2\alpha -si{n}^2\alpha$ / $2si{n}^2\alpha -co{s}^2\alpha$, we will first expand the cosine term using the double angle formula:

cos 2α = cos^2α - sin^2α

Therefore, we can rewrite the expression as:

$co{s}^2\alpha -si{n}^2\alpha -si{n}^2\alpha$ / $2si{n}^2\alpha -co{s}^2\alpha$ $co{s}^2\alpha -2si{n}^2\alpha$ / $2si{n}^2\alpha -co{s}^2\alpha$

Next, we can factor out a negative sign from the numerator:

-$2si{n}^2\alpha -co{s}^2\alpha$ / $2si{n}^2\alpha -co{s}^2\alpha$

Finally, we can cancel out the common terms in the numerator and the denominator:

-1

Therefore, the simplified form of $cos2\alpha -si{n}^2\alpha$ / $2si{n}^2\alpha -co{s}^2\alpha$ is -1.