Where did I go wrong in solving this trigonometry equation?

1 + sec Θ = tan Θ (all solutions on [0,2pi))

1 + sec Θ = sin/cos Θ (tan is the same as sin/cos)

cos Θ + 1 = sin Θ (Multiplied both sides by cos, cos*sec = 1)

cos^2 Θ + 2cos Θ + 1 = sin^2 Θ (Squared both sides)

cos^2 Θ + 2cos Θ + 1 = (1 - cos^2 Θ) (Pythagorean identity for sin^2 Θ)

cos^2 Θ + 2cos Θ = - cos^2 Θ (Took away 1 from both sides)

2cos^2 Θ + 2cos Θ = 0 (Added cos^2 Θ to both sides)

2cos^2 Θ = -2cos Θ (Moved 2cos Θ to the right side)

cos^2 Θ = -cos Θ (Divided both sides by 2)

cos^2 Θ/cos Θ = -1 (Divide both sides by cos Θ)

cos Θ = -1 (Rules of exponents)

I must've done a bunch of unnecessary stuff, but there's so many ways to write trigonometry expressions. I skipped to the answer page of my workbook and it said the answer was 0. I guess I'm wrong because arccos(-1) does not equal 0. Can someone help me out?