Question

How do you factor the expression and use the fundamental identities to simplify `sec^4x-tan^4x`?

Answer 1

`(1 + sin^2 x)/(cos^2 x)`

Explanation:

`S = sec^4 x - tan^4 x = (sec^2 x - tan^2 x)(sec^2 x + tan^2 x)`
The first factor is equal to 1 -->
`sec^2 x - tan^2 x = 1/(cos^2 x) - sin^2 x/(cos^2 x) =`
`= (1 - sin^2 x)/(cos^2 x) = 1`
Develop the second factor -->
`1/(cos^2 x) + sin^2 x/(cos^2 x) = (1 + sin^2 x)/(cos^2 x)`
Finally,
`S = (1 + sin^2 x)/(cos^2 x)`