Find the value of x in each of the following :

We have,

$\tan x=\sin 45^{\circ} \cos 45^{\circ}+\sin 30^{\circ} \ldots \ldots$ (1)

Now we know that

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

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

$\tan x=\sin 45^{\circ} \cos 45^{\circ}+\sin 30^{\circ}$

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

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

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

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

$=\frac{2}{2}$

$=1$

Therefore,

$\tan x=1 \ldots \ldots$ (2)

Since,

$\tan 45^{\circ}=1 \ldots \ldots(3)$

Therefore by comparing equation (2) and (3)

We get,

$x=45^{\circ}$

Therefore,

$x=45^{\circ}$