Final answer:
To demonstrate the equation PQ² = 2 - 2(cosα cosβ + sinα sinβ), we locate points P and Q with coordinates based on their trigonometric relations, apply the distance formula, and simplify using the Pythagorean identity to reach the result.
Step-by-step explanation:
To show that PQ² = 2 - 2(cosα cosβ + sinα sinβ) using the distance formula, we begin by considering two points P and Q on the terminal sides of the angles α and β, respectively. The coordinates of P can be expressed as (cosα, sinα) and those of Q as (cosβ, sinβ), based on the unit circle definitions of sine and cosine.
Using the distance formula:
PQ² = (cosα - cosβ)² + (sinα - sinβ)²
Expanding this equation, we get:
PQ² = cos²α - 2cosαcosβ + cos²β + sin²α - 2sinαsinβ + sin²β
Combining like terms and knowing that sin²α + cos²α = 1 (from the Pythagorean identity), we simplify the equation to:
PQ² = 2 - 2(cosαcosβ + sinαsinβ), which is the desired result.