Prove the identity: sec^4x - tan^4x = 1+2tan^2x

From left side:

sec^4x - tan^4x

factors into:

(sec(x)+tan(x))*(sec(x)-tan(x))*(sec^2(x) + tan^2(x))

(sec(x)+tan(x))*(sec(x)-tan(x)) =sec^2x - tan^2x

and from trig identity:
sec^2x - tan^2x = 1

left side:
1*(sec^2(x) + tan^2(x))

Right side:

1+2tan^2(x)

from the trig identity:
sec^2x - tan^2x = 1

sec^2x - tan^2x + 2tan^2x = 1+2tan^2x

simp lying this:

sec^2x + tan^2x

So right side now matches left side.

Steps r not clear