Evaluate each of the following

We have,

$\sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ}$....(1)

Now $\sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}, \sin 30^{\circ}=\frac{1}{2}, \cos 30^{\circ}=\frac{\sqrt{3}}{2}$

So by substituting above values in equation (1)

We get,

$\sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ}$

$=\frac{1}{\sqrt{2}} \times \frac{1}{2}+\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}$

$=\frac{1}{2 \sqrt{2}}+\frac{\sqrt{3}}{2 \sqrt{2}}$

$=\frac{1+\sqrt{3}}{2 \sqrt{2}}$

Therefore,

$\sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ}=\frac{1+\sqrt{3}}{2 \sqrt{2}}$