Is there another way to solve this trigonometric identity?

Verify the identity.
(1 + tan^2u)(1 - sin^2u) = 1

I solved it with (1 + tan^2u)(1 - sin^2u) = (sec^2u)(cos^2u) = 1/cos^2u(cos^2u) = 1
Are there other ways to solve it?

There are other ways, but they're all pretty much the same (in the same manner that sin² x + cos²x = 1 and 1 + tan² x = sec² x are basically the same identity).
For instance, convert the tan to sin / cos:
(1 + tan² u) (1 - sin² u)
= (1 + (sin² u / cos² u)) (1 - sin² u)
= ((cos² u + sin² u) / cos² u) (1 - sin² u)
= (1 / cos² u) (cos² u)
= 1

Or, probably the simplest, the same argument but applying 1 - sin² u = cos² u earlier:
(1 + tan² u) (1 - sin² u)
= (1 + (sin² u / cos² u)) (cos² u)
= cos² u + sin² u
= 1.