Love these trig proofs! You must get a common denominator with the terms on the left. Fortunately, they happen to be a difference of squares, so {(cosx)(1-sin x) + (cos x)(1 + sin x)}/{(1 + sin x)(1 - sin x)}. The denominator FOILs to 1 - (sin x)^2, which simplifies to (cos x)^2. The numerator factors out a cos x and is then left with 1 + sin x + 1 - sin x. The sin x terms cancel, leaving 2 cos x in the numerator. The cos x in the numerator cancels one of the cos x in the denominator and 1/cos x is sec x. Combine the sec x with the remaining 2 and you have the 2sec x.