Evaluate each of the following:$ \sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ} $

Given:

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

To do:

We have to evaluate \( \sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ} \).

Solution:  

We know that,

$\sin 45^{\circ}=\frac{1}{\sqrt2}$

$\sin 30^{\circ}=\frac{1}{2}$

$\cos 45^{\circ}=\frac{1}{\sqrt2}$

$\cos 30^{\circ}=\frac{\sqrt3}{2}$

Therefore,

$ \sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ}=\frac{1}{\sqrt2}\times\frac{1}{2}+\frac{1}{\sqrt2}\times\frac{\sqrt3}{2}$

$=\frac{1}{2\sqrt2}+\frac{\sqrt3}{2\sqrt2}$

$=\frac{1+\sqrt3}{2\sqrt2}$

Hence, $ \sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ}=\frac{1+\sqrt3}{2\sqrt2}$.