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