1-tan^2 theta/1+tan^2 theta + 2sin^2 theta need to simply to sin and cos

Answer 1

I have a strong feeling you meant:

(1-tan^2 Ø)/(1 + tan^2Ø) + 2tan^2 Ø
= (1 - sin^2 Ø/cos^2 Ø)/sec^2 Ø + 2sin^2 Ø/cos^2 Ø
= (1 - sin^2 Ø/cos^2 Ø)(cos^2 Ø) + 2sin^2 Ø/cos^2 Ø
= cos^2 Ø - sin^2 Ø + 2sin^2 Ø/cos^2 Ø
this is in terms of sines and cosines

we could write it as

cos (2Ø) - 2sin^2 Ø/cos^2 Ø

anything beyond that would only make it more complicated.