I would recommend multiplying the equation by the product of the two denominators, cosθ(1-sinθ), which will immediately get the fractions out of the picture.
(1+sinθ)/cosθ = cosθ/(1-sinθ)
cosθ(1-sinθ)(1+sinθ)/cosθ = cosθ(1-sinθ)cosθ/(1-sinθ)
which allows us to factor out common terms on each side
(1-sinθ)(1+sinθ) = cosθ cosθ
which is easily rewritten
1 - sin²θ = cos²θ
Isolate the 1 by adding sin²θ to both sides
1 = sin²θ + cos²θ
which is is the fundamental trigonometric identity. Since we know this to be true, we know the original equation to be true.