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  4. If sin2 θ + sin2 φ = 1/2, cos2 + cos2 φ = 3/2, then cos2 (θ - φ) is equal to

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Q. If $\sin^2 \theta + \sin^2 \phi = 1/2, \cos^2 + \cos^2 \phi = 3/2$, then $\cos^2 (\theta - \phi)$ is equal to

BITSATBITSAT 2012

Solution:

Using cosine formula
$2 \sin (\theta + \phi) \cos (\theta - \phi) = 1/2$ ......(i)
$2 \cos (\theta + \phi) \cos (\theta - \phi) = 3/2$ .....(ii)
Squaring (1) and (2) and then adding
$4 \cos^{2} \left(\theta - \phi\right) = \frac{1}{4} + \frac{9}{4} = \frac{5}{2}$
$ \Rightarrow \cos^{2} \left(\theta -\phi\right) = \frac{5}{8} $