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Air $\displaystyle \tan \left(\alpha+\beta\right) = \frac{\tan \alpha+\tan \beta}{1-\tan\alpha\tan\beta}$
$\displaystyle \frac{19\pi}{12} = \frac{4\pi}{3} + \frac{\pi}{4}$
$\displaystyle \therefore \tan \left( \frac{19 \pi}{12} \right) = \tan \left(\frac{4\pi}{3} + \frac{\pi}{4}\right)$
$\displaystyle =\frac{\tan
\left(\frac{4\pi}{3}\right)
+\tan \left(\frac{\pi}{4}\right)}{1-\tan
\left(\frac{4\pi}{3}\right)
\tan \left(\frac{\pi}{4}\right)} = \frac{\sqrt{3}+1}{1-\sqrt{3}} $
To simplify $\displaystyle \frac{\sqrt{3}+1}{1-\sqrt{3}} $, multiply numerator and the denominator by the conjugate of the denominator which is $\displaystyle 1+\sqrt{3}$.