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I figured this out using my calculator but I need help solving through it without. 
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2 Answers

Calculators only give approximate answers, no matter how many decimals you get from it. They can serve as guides, but only an analytic method will get exact answers.
Let's use the alternative technique in:
cos(θ)/(1+sin(θ))+tan(θ) =
(x/r)/(1 + y/r) + y/x =
x/(r + y) + y/x =
x*x/(x*(r + y)) + y*(r + y)/(x*(r + y)) =
(x*x + y*(r + y))/(x*(r + y)) =
(x^2 + yr + y^2)/(x*(r + y)) =
(r^2 + yr)/(x*(r + y)) =
r(r + y)/(x*(r + y)) =
r/x = 1/cos(θ) = sec(θ).
we know that: tanX=sinX/cosX, therefore:
common denominator would be cosX(1+sinX), so the addition of these two terms would be:
knowing sinX^2+cosX^2=1, we'll have:
cancel out (1+sinX)
the answer would be: