To prove the triangle inequality for the given metric

Question

I need to prove that the metric defined by

$d(x,y)=\frac{2|x-y|}{\sqrt{1+|x^2|}\sqrt{1+|y^2|}}$

forms a metric on the set of all complex numbers.
I am facing problem with the denominator in proving the triangle inequality as the term in the denominator is under square root and there is also modulus inside it. Need help in proving this.

Thanks!