Evaluate each of the following

We have to find the value of the following expression

$\cos 60^{\circ} \cos 45^{\circ}-\sin 60^{\circ} \sin 45^{\circ}$....(1)

Now $\sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}, \sin 60^{\circ}=\frac{\sqrt{3}}{2}, \cos 60^{\circ}=\frac{1}{2}$

So by substituting above values in equation (1)

We get,

$\cos 60^{\circ} \cos 45^{\circ}-\sin 60^{\circ} \sin 45^{\circ}$

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

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

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

Therefore,

$\cos 60^{\circ} \cos 45^{\circ}-\sin 60^{\circ} \sin 45^{\circ}=\frac{1-\sqrt{3}}{2 \sqrt{2}}$