Question

Find the value of $\tan \frac{13\pi }{12}$

Answer 1

$\tan \frac{13\pi }{12}$

$\tan (\frac{13\times 180}{12})$

$\tan 195°$

$\tan \left(180°-15°\right)$

$\tan \left(180°+\theta \right)=\tan \theta$

$\tan 15°=\tan (45°-30°)$

$\tan (A-B)$ $=\frac{\tan A-\tan B}{1+\tan A\tan B}$

$=\frac{\tan 45-\tan 30}{1+\tan 45\tan 30}$

$=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}$

$=\frac{{(1-\frac{1}{\sqrt{3}})}^2}{1-\frac{1}{3}}$

$=\frac{1+\frac{1}{3}-\frac{2}{\sqrt{3}}}{\frac{2}{3}}$

$=\frac{\frac{3+1-2\sqrt{3}}{3}}{\frac{2}{3}}$

$=2\sqrt{3}$