Question

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

Answer 1

$\begin{array}{c}\tan \frac{13\pi }{12}=\tan \left(\frac{13\times 180}{12}\right)\\ =\tan \left(\pi +15\right)\\ =\tan \left({15}^0\right)\\ =\tan \left({60}^0-{45}^0\right)\\ =\frac{\tan \left({60}^0\right)-\tan \left({45}^0\right)}{1+\tan \left({60}^0\right)\tan \left({45}^0\right)}\\ =\frac{\sqrt{3}-1}{1+\sqrt{3}}\\ =\frac{\sqrt{3}-1}{1+\sqrt{3}}\times \frac{\sqrt{3}-1}{\sqrt{3}-1}\\ =\frac{{\left(\sqrt{3}-1\right)}^2}{3-1}\\ =\frac{3+1-2\sqrt{3}}{2}\\ =\frac{4-2\sqrt{3}}{2}=2-\sqrt{3}\end{array}$