Answer
The required value is $2-\sqrt{3}$
Work Step by Step
We know that the identity $\tan \left( \theta -\phi \right)$ is the difference between the tangent of the first angle and the tangent of the second angle divided by 1 plus the product of both angles.
$\tan \left( \theta -\phi \right)=\frac{\tan \theta -\tan \phi }{1+\tan \theta \tan \left( \phi \right)}$
We use the above-mentioned identity where $\theta =\frac{4\pi }{3}$ and $\phi =\frac{\pi }{4}$
$\begin{align}
& \tan \left( \frac{4\pi }{3}-\frac{\pi }{4} \right)=\frac{\tan \frac{4\pi }{3}-\tan \frac{\pi }{4}}{1+\tan \frac{4\pi }{3}.\tan \frac{\pi }{4}} \\
& =\frac{\sqrt{3}-1}{1+\sqrt{3}.1} \\
& =\frac{\sqrt{3}-1}{1+\sqrt{3}}.\frac{1-\sqrt{3}}{1-\sqrt{3}} \\
& =\frac{-{{\left( 1-\sqrt{3} \right)}^{2}}}{1-3}
\end{align}$
And the above expression can be simplified by applying the algebraic identity ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$
$\begin{align}
& \frac{-{{\left( 1-\sqrt{3} \right)}^{2}}}{1-3}=\frac{-\left( 1-2.\sqrt{3}+3 \right)}{-2} \\
& =\frac{-\left( 4-2\sqrt{3} \right)}{-2} \\
& =\frac{-2\left( 2-\sqrt{3} \right)}{-2} \\
& =2-\sqrt{3}
\end{align}$
Hence, the required value is $2-\sqrt{3}$
