sick of these identities?

Question

tan^4(x) +2tan^2(x) +1

Factor and explain. Thanks

Answer 1

Let u=tan(x). Then, we're factoring

u^4 + 2u^2 + 1
= (u²+1)²
= (tan²(x) + 1)²
= (sec²(x))²
= sec^4(x)

This makes use of the identity tan²(x)+1 = sec²(x).

That's about as reduced a form as you're going to get.

Good luck, work hard, and stay away from drugs.

Answer 2

tan^4(x) +2tan^2(x) +1 = (tan^2(x) + 1) (tan^2(x) + 1)

Answer 3

= (tan ² x + 1).(tan ² x + 1)
= sec ² x . sec ² x
= sec^(4) x