My bad, I didn't check that last step correctly.
This may not quite be within the MX2 syllabus, but if you differentiate the expression
\frac{1 - a^2b^2}{(1+a^2)(1+b^2)} + \frac {a+b}{\sqrt{(1+a^2)(1+b^2)}}
twice, as a function of a and then as a function of b , it can be shown that...