[1 + sin(theta) + cos(theta)] / [1 + sin(theta) - cos(theta)] = [1+cos(theta)]/ sin(theta)

Answer 1

Suggestion:
On LS, multiply top and bottom by (1 + sinθ + cosθ)/(1 + sinθ + cosθ)
Be careful, you will have 9 terms on the top and 9 at the bottom

On the top you will have among them sin^2 θ + cos^2 θ, replace that with 1 to get
on top: 2 + 2sinθ + 2cosθ + 2sinθcosθ
group it and factor the top to 2(1+sinθ)(1+cosθ)

on the bottom: replace 1 with sin2 θ + cos^2 θ, that way the +cos^2 θ will cancel the -cos^2 θ at the end
other terms will cancel as well, leaving you with
= 2sin^2 θ + 2sinθ
= 2sin(sinθ + 1)
Well, well, will you look at that!!!

Answer 2

Thanks for teaching me how to solve my trig problems. I had trouble with those two and your explanations guided me how to solve these types of problems. :)