I) sin^2 A sec^2 B + tan^2 B cos^2 A = sin^2A + tan²B

sin² A sec² B + tan² B cos² A

A good first step is to write everything in terms of sine and cosine.

sin² A / cos² B + sin² B cos² A / cos² B

The fractions have the same denominator, so combine into one:

(sin² A + sin² B cos² A) / cos² B

Using Pythagorean identity, we can rewrite sin² B as 1 - cos² B:

(sin² A + (1 - cos² B) cos² A) / cos² B

Distribute:

(sin² A + cos² A - cos² B cos² A) / cos² B

Pythagorean identity:

(1 - cos² B cos² A) / cos² B

Now divide into two fractions again:

1 / cos² B - cos² B cos² A / cos² B

Simplify:

sec² B - cos² A

Using Pythagorean identity again:

(tan² B + 1) - (1 - sin² A)

tan² B + 1 - 1 + sin² A

tan² B + sin² A