sin^2x (sec^2x + csc^2x) = sec^2x
I would convert the functions in the parentheses to their reciprocals.
sin^2x (1/cos^2x + 1/sin^2x) = sec^2x
Now distribute the sine.
sin^2x/cos^2x + sin^2x/sin^2x = sec^2x
Remember that sine divided by cosine is always tangent.
tan^2x + sin^2x/sin^2x = sec^2x
The remaining fraction is simply 1.
tan^2x + 1 = sec^2x
Use the Pythagorean identity to add the left side.
sec^2x = sec^2x
Q.E.D.