Question

If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).

Answer 1

$$\begin{gathered} \cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\ =\frac{2\cos \alpha \cos \beta -2\sin \alpha \sin \beta +2-2}{2} \end{gathered}$$
$=\frac{{\sin }^2\alpha +{\cos }^2\alpha +{\sin }^2\beta +{\cos }^2\beta +2\cos \alpha \cos \beta -2\sin \alpha \sin \beta -2}{2}$
$=\frac{(sin\alpha -s\mathrm{in}\beta {)}^2+(\cos \alpha +\cos \beta {)}^2-2}{2}$
$=\frac{a^2+b^2-2}{2}$