The expression (tanθ + secθ − 1)/(tanθ − secθ + 1) simplifies to:

Replace the −1 in the numerator by se$c^2$θ − ta$n^2$θ = (secθ − tanθ)(secθ + tanθ). The numerator becomes (secθ + tanθ)(1 − secθ + tanθ), which is (secθ + tanθ) times the denominator, leaving secθ + tanθ. Check at θ = 60°: (1.732 + 2 − 1)/(1.732 − 2 + 1) = 2.732/0.732 = 3.732 = sec60° + tan60°.