A student writes: 'Since si$n^2$θ + co$s^2$θ = 1, taking the square root of both sides gives sinθ + cosθ = 1 for every θ.' Which statement correctly identifies the faulty step?

The single invalid move is rooting a sum term by term — in general √$(p^2+q^2)$ ≠ $p+q$. At θ = 45° the true value of sinθ + cosθ is √2 ≈ 1.414, not 1, which kills the claim outright. The Pythagorean identity itself is perfectly true for every angle, so the first and fourth statements are false, and writing ±1 on the right does not rescue anything, because the error is on the LEFT side of the equation.