If sin α − sin β = a and cos α + cos β = b, Find cos(α + β)
Question:
If
\[
\sin\alpha-\sin\beta=a
\]
and
\[
\cos\alpha+\cos\beta=b
\]
find
\[
\cos(\alpha+\beta)
\]
Solution
Using, \[ \sin\alpha-\sin\beta = 2\cos\frac{\alpha+\beta}{2} \sin\frac{\alpha-\beta}{2} \]
\[ a = 2\cos\frac{\alpha+\beta}{2} \sin\frac{\alpha-\beta}{2} \]
Also, \[ \cos\alpha+\cos\beta = 2\cos\frac{\alpha+\beta}{2} \cos\frac{\alpha-\beta}{2} \]
\[ b = 2\cos\frac{\alpha+\beta}{2} \cos\frac{\alpha-\beta}{2} \]
Squaring and adding, \[ a^2+b^2 = 4\cos^2\frac{\alpha+\beta}{2} \]
\[ \cos^2\frac{\alpha+\beta}{2} = \frac{a^2+b^2}{4} \]
Using, \[ \cos\theta=2\cos^2\frac{\theta}{2}-1 \]
\[ \cos(\alpha+\beta) = 2\left(\frac{a^2+b^2}{4}\right)-1 \]
\[ \boxed{ \cos(\alpha+\beta) = \frac{a^2+b^2}{2}-1 } \]