Proving sec⁶x - tan⁶x = 1 + 3sec²x tan²x

[happyg1]

Homework Statement

Show that the LHS can be changed into the RHS.
$$sec^6 x-tan^6x=1+3sec^2x tan^2x$$

Homework Equations

Trig identities.

The Attempt at a Solution

I tried factoring the LHS:
$$(sec^2-tan^2)(sec^4+sec^2tan^2+tan^4)$$
$$sec^2-tan^2=1$$ so that leaves me with the other thing in the parentheses. I have tried using the Pythagorean identities on sec^2 and tan^2, I have broken up the sec^4 into sec^2*sec^2...

I just am not getting anywhere.
Pointers would be nice.
CC

 

[icystrike]

i get the RHS le.
try (tan²x+1)³ - tan^6 x
then you expand.
Cheers

i will be offline , i shall leave the ans in the spoiler (:

next step:

Spoiler

3tan²x+3tan^4 x +1

next:

Spoiler

1+3tan²x(tan²x+1)

Finally:
1+3sec²tan²x
=RHS(shown)

 

[happyg1]

I got it easily after the first hint. Thanks! I had been staring at it too long to see another way.