Verify the identity:

(1+tan^2u)(1-sin^2u)=1

=> The first expression on the left:  (1 + tan²)

we know that tangent = sin/cos
so tan² = sin²/cos²
the expression is  (1 + sin²/cos²)

=>  The second expression on the left:  (1 - sin²)

we know that (sin² + cos²) = 1
subtract sin² from each side,
and we have    (1 - sin²) = (cos²)

=>  Multiply the massaged form of the two expressions:

(1 + tan²) (1 - sin²)  =   (1 + sin²/cos²) (cos²) = (cos² + sin²)  =  1       qed