Can the following identity be verified: sec^4 x - sec^2 x = tan^4 x + tan^2 x?

We need to verify that (sec x)^4 - (sec x)^2 = (tan x)^4 +
(tan x)^2.

We know that (sin x)^2 + (cos x)^2 =
1

=> (sin x)^2 / (cos x)^2 + (cos x)^2/ (cos x)^2 =
1/(cos x)^2

=> (tan x)^2 + 1 = (sec
x)^2

Starting with the left hand
side:

(sec x)^4 - (sec
x)^2

=> (sec x)^2[(sec x)^2 -
1]

=> [(tan x)^2 + 1][(tan x)^2 + 1 -
1]

=> [(tan x)^2 + 1][(tan
x)^2]

=> (tan x)^4 + (tan
x)^2

which is the right hand
side.

This proves that (sec x)^4 - (sec x)^2
= (tan x)^4 + (tan x)^2.