Solving an Equation for y in terms of x How is it possible to solve the equation:
$$ x = \frac{2-y}{y} \sqrt{1-y^2} $$
for $ y $ in terms of $ x $?
I know that it is possible to do it, since WolframAlpha gives a complicated answer but entirely composed of elementary functions. However, I am stumped on how to even start.
Thanks in advance!

for the $\sqrt{1-y^2}$ to make sense we need  $-1 \le y \le 1.$ therefore we can set $ y = \cos t.$ with that, we have $$x = \frac{2-\cos t}{\cos t}\sqrt{1- \cos^2 t} = \frac{(2 - \cos t)\sin t}{\cos t}, 0 \le t \le \pi.$$  we can solve $$\frac{(2 - \cos t)\sin t}{\cos t} = x \tag 1$$ for $$\sin t = \frac{x\cos t}{2 - \cos t}\to (2 - \cos t)^2\cos^2 t + x^2\cos^2t = 1 $$ that is $$
cos^4t-4\cos^3 t+(x^2+4)\cos^2t-1 = 0, y = \cos t.$$
