Question

Prove

 

sec ^4 (x) - 2 sec ^2 (x) tan ^2 n (x) + tan ^4 (x)=1

Answer 1

this seems to be a perfect square. You can factor this.
Observe:
(sec^2x-tan^2x)(sec^2x-tan^2x)=1 (if you FOIL this, you get your starting equation)

This is a variation of the pythagorean identity
1+tan^2x=sec^2x.
1=sec^2x-tan^2x <-- this is the same as the parenthesis

Simplifying the parenthesis, you get 1 times 1, which equals 1, thus proving your equation.
Tada!! Hope this helped. :)