Find the value of x in each of the following :

We have,

$\cos 2 x=\cos 60^{\circ} \cos 30^{\circ}+\sin 60^{\circ} \sin 30^{\circ}$

Now we know that

$\sin 60^{\circ}=\cos 30^{\circ}=\frac{\sqrt{3}}{2}$ and $\sin 30^{\circ}=\cos 60^{\circ}=\frac{1}{2}$

Now by substituting above values in equation (1), we get,

$\cos 2 x=\cos 60^{\circ} \cos 30^{\circ}+\sin 60^{\circ} \sin 30^{\circ}$

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

$=\frac{\sqrt{3}}{4}+\frac{\sqrt{3}}{4}$

$=\frac{2 \sqrt{3}}{4}$

Therefore,

$\cos 2 x=\frac{2 \sqrt{3}}{4}$

Now $\frac{2 \sqrt{3}}{4}$ gets reduced to $\frac{\sqrt{3}}{2}$

Therefore,

$\cos 2 x=\frac{\sqrt{3}}{2} \ldots \ldots(2)$

Since,

$\cos 30^{\circ}=\frac{\sqrt{3}}{2}$ …… (3)

Therefore by comparing equation (2) and (3)

We get,

$2 x=30^{\circ}$

$\Rightarrow x=\frac{30^{\circ}}{2}$

$\Rightarrow x=15^{\circ}$

Therefore,

$x=15^{\circ}$