Question:
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
Solution:
$\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta$
$=\frac{2 \cos \alpha \cos \beta-2 \sin \alpha \sin \beta+2-2}{2}$
$=\frac{\sin ^{2} \alpha+\cos ^{2} \alpha+\sin ^{2} \beta+\cos ^{2} \beta+2 \cos \alpha \cos \beta-2 \sin \alpha \sin \beta-2}{2}$
$=\frac{(\sin \alpha-s \text { in } \beta)^{2}+(\cos \alpha+\cos \beta)^{2}-2}{2}$
$=\frac{a^{2}+b^{2}-2}{2}$
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