I) sin^2 a sec^2 b + tan^2 b cos^2 a = sin^2a + tan²b

Explanation:

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

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helium

oxygen

neon

sodium