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Find the value of $ x $ in each of the following:$ \tan x=\sin 45^{\circ} \cos 45^{\circ}+\sin 30^{\circ} $
Given:
\( \tan x=\sin 45^{\circ} \cos 45^{\circ}+\sin 30^{\circ} \)
To do:
We have to find the value of \( x \).
Solution:
We know that,
$\sin 30^{\circ}=\frac{1}{2}$
$\sin 45^{\circ}=\frac{1}{\sqrt2}$
$\cos 45^{\circ}=\frac{1}{\sqrt2}$
Therefore,
\( \tan x=\sin 45^{\circ} \cos 45^{\circ}+\sin 30^{\circ} \)
$\Rightarrow \tan x=\frac{1}{\sqrt{2}}\times\frac{1}{\sqrt{2}}+\frac{1}{2}$
$\Rightarrow \tan x=\frac{1}{2}+\frac{1}{2}$
$\Rightarrow \tan x=1$
$\Rightarrow \tan x=\tan 45^{\circ}$ (Since $\tan 45^{\circ}=1$)
Comparing on both sides, we get,
$x=45^{\circ}$
Hence, the value of $x$ is $45^{\circ}$.
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