Question

Simplify:
1) 1 + sin²α + cos²α

2) cos²α + tan²α*cos²α

Answer 1

Looks like you need to review your most fundamental of all trig identities:

sin²α + cos²α = 1

divide that through by cos²α and you get
tan²α + 1 = sec²α

The rest is just Algebra I

Answer 2

1)
Trig identity: sin²α = 1 - cos²α
1 + 1 - cos²α + cos²α = 2

2)
Trig identity: 1 + tan²α = sec²α, sec²α = 1/cos²α
cos²α(1 + tan²α) = cos²α(sec²α)
= 1