Triangles AOC and BOD are congruent by SAS. Therefore, and these lengths can be found using the . Because this is the unit circle, the coordinates of points A, B, and C are equal to cosine and sine of , – β, and β respectively. Write expressions for the lengths, set them equal, and square both sides of the equation. Simplifying the resulting expressions involving cosine and sine of , β, and – β requires using the identity. When simplified, the equation becomes cos( – β) = cos()cos(β) + sin()sin(
Squaring the equal lengths produces terms involving squared sine and cosine values. To simplify them, use the Pythagorean identity, sin²θ + cos²θ = 1. This leads to the cosine difference formula, cos(α − β) = cosα cosβ + sinα sinβ, so the needed identity is Pythagorean.
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