Worked example

Solve tan 2x = 1 on [0, π)

Answer

x = π/8 and x = 5π/8

tan 2x = 1
Direct route: 2x = π/4 + kπ.
x = π/8 + kπ/2
Divide by 2 — so x = π/8, 5π/8 on [0, π).
Alternative: expand with the double-angle formula
2 tan x/(1 − tan²x) = 1. · Double-angle identity for tangent
tan²x + 2 tan x − 1 = 0 → tan x = −1 ± √2
Quadratic in tan x; the two roots are exactly tan(π/8) and tan(5π/8) — a good check that both routes agree.

tan 2x = 2 tan x / (1 − tan²x)

Double-angle identity for tangent — Everything in the problem is a tangent — solving tan 2x = 1, or rewriting a slope-doubling situation. Also the formula that shows why tan has period 180° but tan 2x compresses it.

Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.