Question

How do you simplify `sec^4x-tan^4x`?

Answer 1

`1 + 2tan^2 x`

Explanation:

`sec^4 x - tan^4 x = (sec^2 x - tan^2 x)(sec^2 x + tan^2 x)`
Since
`(sec^2 x - tan^2 x) = [(1/(cos^2 x) - (sin^2 x/(cos^2 x))] =`
`= (1 - sin^2 x)/(cos^2 x) = cos^2 x/(cos^2 x) = 1`,
there for;
`sec^4x - tan^4 x = sec^2 x + tan^2 x`
Reminder: `sec^2 x = (1 + tan^2 x)`
Finally,
`sec^4 x - tan^4 x = 1 + 2tan^2 x`