Question

If sin α + sin β = a and cos α + cos β = b, then prove that cos(α – β) = `(a^2 + b^2 - 2)/2`

Answer 1

Consider a² + b² = sin²α + sin²β + 2 sin α sin β + cos²α + cos²β + 2 cos α cos β

a² + b² = (sin²α + cos²α) + (sin²β + cos²β) + 2[cos α cos β + sin α sin β]

a² + b² = 1 + 1 + 2 cos(α – β)

∴ cos(α – β) = `(a^2 + b^2 - 2)/2`