Prove that : sin2 θ + cos2 θ = 1.

Proof:

Take a circle with centre at the origin O and radius r. Let the circle cut the positive X-axis at P(r, 0). Take a point Q(x, y) on the circle, let ∠POQ = θ and draw QA ⊥ OX, so OA = x and QA = y.

In right-angled Δ OAQ, OQ = r (radius), so by the Pythagoras theorem,

OA² + QA² = OQ², i.e. x² + y² = r².

Dividing both sides by r²: \(\left(\dfrac{x}{r}\right)^2 + \left(\dfrac{y}{r}\right)^2 = 1\).

By definition, \(\sin\theta = \dfrac{y}{r}\) and \(\cos\theta = \dfrac{x}{r}\), hence

cos² θ + sin² θ = 1.