Question

How do you simplify `sec^3x-sec^2x-secx+1`?

Answer 1

Simplify: sec^3 x - sec^2 x - sec x + 1

Explanation:

Call sec x = t, we get:
`f(t) = t^3 - t^2 - t + 1`
Since (a + b + c + d) = 0, one factor is (t - 1)
Since a - b + c - d = 0, one factor is (t - 1)
Finally `f(t) = (t - 1)^2(t + 1)`
`f(x) = (sec x - 1)^2(sec x + 1)`

Answer 2

`sec^3 x - sec^2 x - sec x + 1 = (sec x - 1)tan^2 x`

Explanation:

`tan^2 A = sec^2 A-1`

`sec^3 x - sec^2 x - sec x + 1 = sec x(sec^2 x - 1) - (sec^2 x - 1) = (sec x - 1)(sec^2 x - 1) = (sec x - 1)tan^2 x`