tan^x-secx=1

looks like a typo

tan^x is meaningless

did you mean tan^2 x ??

I will assume that is what you meant ...

sin^2 x/cos^2 x - 1/cosx = 1
times cos^2 x
sin^2 x - cosx = cos^2 x
1 - cos^2 x - cosx = cos^2 x
2cos^2 x + cosx - 1 = 0
(2cosx - 1)(cosx + 1) = 0
cosx = 1/2 or cosx = -1
x = 60° or 300° or x = 180°
x = π/3 , 5π/3, or π