If $(\cos \alpha+\cos \beta)^{2}+(\sin \alpha+\sin \beta)^{2}=\lambda \cos ^{2}\left(\frac{\alpha-\beta}{2}\right)$, write the value of $\lambda$
$(\cos \alpha+\cos \beta)^{2}+(\sin \alpha+\sin \beta)^{2}=\lambda \cos ^{2}\left(\frac{\alpha-\beta}{2}\right)$
Consider LHS:
(cos α + cos β)2 + (sin α + sin β)2
$=\left[2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)\right]^{2}+\left[2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)\right]^{2}$
$=4 \cos ^{2}\left(\frac{\alpha+\beta}{2}\right) \cos ^{2}\left(\frac{\alpha-\beta}{2}\right)+4 \sin ^{2}\left(\frac{\alpha+\beta}{2}\right) \cos ^{2}\left(\frac{\alpha-\beta}{2}\right)$
$=4 \cos ^{2}\left(\frac{\alpha-\beta}{2}\right)\left[\cos ^{2}\left(\frac{\alpha+\beta}{2}\right)+\sin ^{2}\left(\frac{\alpha+\beta}{2}\right)\right]$
$=4 \cos ^{2}\left(\frac{\alpha-\beta}{2}\right)$
= RHS
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