Answer
The exact value of $\tan \left( \frac{5\pi }{3}-\frac{\pi }{4} \right)$ is $2+\sqrt{3}$.
Work Step by Step
Use the difference formula of the tangent and evaluate the expression as,
$\tan \left( \frac{5\pi }{3}-\frac{\pi }{4} \right)=\frac{\tan \frac{5\pi }{3}-\tan \frac{\pi }{4}}{1+\tan \frac{5\pi }{3}\tan \frac{\pi }{4}}$
Substitute the values, $\tan \frac{5\pi }{3}=-\sqrt{3}$ and $\tan \frac{\pi }{4}=1$.
$\begin{align}
& \tan \left( \frac{5\pi }{3}+\frac{\pi }{4} \right)=\frac{-\sqrt{3}-1}{1+\left( -\sqrt{3}\times 1 \right)} \\
& =-\left( \frac{1+\sqrt{3}}{1-\sqrt{3}} \right)
\end{align}$
Further, rationalize the result and solve as,
$\begin{align}
& \tan \left( \frac{5\pi }{3}-\frac{\pi }{4} \right)=-\left( \frac{1+\sqrt{3}}{1-\sqrt{3}} \right)\times \frac{1+\sqrt{3}}{1+\sqrt{3}} \\
& =-\frac{{{\left( 1+\sqrt{3} \right)}^{2}}}{{{1}^{2}}-{{\left( \sqrt{3} \right)}^{2}}} \\
& =-\frac{\left( 1+3+2\sqrt{3} \right)}{1-3} \\
& =\frac{4+2\sqrt{3}}{2}
\end{align}$
Thus,
$\begin{align}
& \tan \left( \frac{5\pi }{3}+\frac{\pi }{4} \right)=\frac{2\left( 2+\sqrt{3} \right)}{2} \\
& =2+\sqrt{3}
\end{align}$
Hence, the exact value of $\tan \left( \frac{5\pi }{3}+\frac{\pi }{4} \right)$ is equivalent to $2+\sqrt{3}$.
