Can you prove sec 2A + tan 2A = tan (45+A)?

I could, I suppose

1 / cos(2a) + sin(2a) / cos(2a) =>
(1 + sin(2a)) / (cos(2a)) =>
(sin(a)^2 + cos(a)^2 + 2 * sin(a) * cos(a)) / (cos(a)^2 - sin(a)^2) =>
(sin(a) + cos(a))^2 / ((cos(a) - sin(a)) * (cos(a) + sin(a))) =>
(cos(a) + sin(a)) / (cos(a) - sin(a)) =>
cos(a) * (1 + sin(a) / cos(a)) / (cos(a) * (1 - sin(a)/cos(a))) =>
(1 + tan(a)) / (1 - tan(a)) =>
(tan(45) + tan(a)) / (1 - tan(a) * tan(45)) =>
tan(a + 45)