Solve exactly over 0 ≤ θ < 2π

Question

secx + tanx = 1

Answer 1

secx + tanx = 1
1 + sinx = cosx
1 + 2sinx + sin^2 x = cos^2 x
1 + 2sinx + sin^2 x = 1 - sin^2 x
2sin^2 x + 2sinx = 0
2sinx(sinx+1)=0
so, sinx=0 or sinx = -1
x=0,π,3π/2

but secx,tanx are undefined at x=3π/2,
secπ + tanπ = -1, not 1, so
x=0 is the only solution