How do I simply $\displaystyle (1+tan^2A)(1-sin^2A)$
Here is another way you could do it,
$\displaystyle 1- sin^2A +tan^2A-tan^2Asin^2A= 1 -sin^2A +\frac{sin^2A}{cos^2A} -\frac{sin^4A}{cos^2A}$
$\displaystyle \frac{cos^2A-sin^2Acos^2A+sin^2A-sin^4A}{cos^2A}$
$\displaystyle \frac{cos^2A(1-sin^2A)+sin^2A(1-sin^2A)}{cos^2A}$
$\displaystyle \frac{cos^4A+sin^2Acos^2A}{cos^2A}$
$\displaystyle \frac {cos^2A(cos^2A + sin^2A)}{cos^2A}$
$\displaystyle cos^2A + sin^2A= 1$