Is -sec(x) +1 the same as cos(x)?
No.
For example, say that x = 30°:
cos(30°) = √3/2 (approximately 0.8660)
sec(x) = 1 / cos(x) ---> sec(30°) = 1 / (√3 / 2) = 2/√3 (approximately 1.1547)
-sec(30°) + 1 = -2/√3 + 1 (approximately -0.1547)
And, this isn't cos(30°).