Given the right triangle shown below, prove cos2(θ) + sin2(θ) = 1. A triangle with sides x, y, and hypotenuse r. Angle theta is opposite to side y.

By
, cos(θ) = StartFraction x Over r EndFraction, and by
, sin(θ) = StartFraction y Over r EndFraction.

Multiplying both sides of the above equations by r, we get that x = r cos(θ) and y = r sin(θ).

The
states that x2 + y2 = r2.

By
, we have [r cos(θ)]2 + [r sin(θ)]2 = r2.

Applying the
, the equation can be written as r2 cos2(θ) + r2 sin2(θ) = r2.

Dividing both sides of the equation by r2 results in cos2(θ) + sin2(θ) = 1.