Question

Verify the identity. Show your work. cos(α - β) - cos(α + β) = 2 sin α sin β

Answer 1

$\bf \textit{Sum and Difference Identities} \\\\ cos(\alpha + \beta)= cos(\alpha)cos(\beta)- sin(\alpha)sin(\beta) \\\\ cos(\alpha - \beta)= cos(\alpha)cos(\beta) + sin(\alpha)sin(\beta)\\\\ -------------------------------$

$\bf {cos(\alpha - \beta)-cos(\alpha+\beta)}=2sin(\alpha)sin(\beta) \\\\\\ \stackrel{\textit{left-hand-side}}{[cos(\alpha)cos(\beta) + sin(\alpha)sin(\beta)]-[cos(\alpha)cos(\beta)- sin(\alpha)sin(\beta)]} \\\\\\ \underline{cos(\alpha)cos(\beta)} + sin(\alpha)sin(\beta)-\underline{cos(\alpha)cos(\beta)} + sin(\alpha)sin(\beta) \\\\\\ sin(\alpha)sin(\beta)+ sin(\alpha)sin(\beta)\implies 2 sin(\alpha)sin(\beta)$