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

Question

If sin α + sin β = a and cos α + cos β = b, then prove that cos(α – β) = \(\frac{a^2+b^2-2}{2}\)

Answer 1

Consider a² + b² = sin² α + sin² β + 2 sin α sin β + cos² α + cos² β + 2 cos α cos β 

a² + b² = (sin² α + cos² α) + (sin² β + cos² β) + 2[cos α cos β + sin α sin β] 

a² + b² = 1 + 1 + 2 cos(α – β) 

∴ cos(α – β) = \(\frac{a^2+b^2-2}{2}\)