A student argues: 'Since ta$n^2$θ = se$c^2$θ − 1, we may write tanθ = √(se$c^2$θ − 1) for every θ. At θ = 120° we have secθ = −2, so tan120° = √(4 − 1) = √3.' In fact tan120° = −√3. The invalid step is:

Every other statement in the chain is sound: cos120° = −1/2 so sec120° = −2 is correct; the identity ta$n^2$θ = se$c^2$θ − 1 holds wherever both are defined, and here both sides equal 3; and tan120° is perfectly well defined, equal to −1.7321. The single false move is 'tanθ = √(ta$n^2$θ)' — the radical always delivers the non-negative root, so it produces |tanθ|. In the second quadrant the tangent is negative, so the correct conclusion is tan120° = −√(se$c^2$120° − 1) = −√3.