The Pythagorean identity might be useful here rather than a double angle identity.
Remember that sin^2 + cos^2 = 1 and so if we divide both sides by cos^2, we get
sin^2/cos^2 + cos^2/cos^2 = 1/cos^2 or
tan^2 + 1 = sec^2
The (tan(v) + 1)^2 ≠ sec^2(v). Rather it is equal to (tan(v) + 1)(tan(v) + 1), which you need to FOIL out; the same goes for the (tan(v) - 1)(tan(v) - 1).