If $\sin 2\theta +\sin 2\varphi =1/2,\cos 2\theta +\cos 2\varphi =3/2,$ then value of ${\cos }^2(\theta -\varphi )$ is

Using cosine formula

$\begin{array}{r}\sin 2\theta +\sin 2\varphi =2\sin (\theta +\varphi )\cos (\theta -\varphi )=1/2\\ \cos 2\theta +\cos 2\varphi =2\cos (\theta +\varphi )\cos (\theta -\varphi )=3/2\end{array}$

Squaring (i) and (ii) and then adding

$\begin{array}{l}4{\cos }^2(\theta -\varphi )=\frac{1}{4}+\frac{9}{4}=\frac{5}{2}\\ \Rightarrow {\cos }^2(\theta -\varphi )=\frac{5}{8}\end{array}$