Finding $\mathrm{cot}(\frac{\pi}{12})$

Given:

$\mathrm{cot}(\theta -\varphi )=\frac{\mathrm{cot}\theta \mathrm{cot}\varphi +1}{\mathrm{cot}\theta -\mathrm{cot}\varphi}$

And:

$\mathrm{cot}\frac{\pi}{3}=\frac{1}{\sqrt{3}};\mathrm{cot}\frac{\pi}{4}=1$

$\frac{\frac{1}{\sqrt{3}}(1)+1}{\frac{1}{\sqrt{3}}-1}$

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

Given:

$\mathrm{cot}(\theta -\varphi )=\frac{\mathrm{cot}\theta \mathrm{cot}\varphi +1}{\mathrm{cot}\theta -\mathrm{cot}\varphi}$

And:

$\mathrm{cot}\frac{\pi}{3}=\frac{1}{\sqrt{3}};\mathrm{cot}\frac{\pi}{4}=1$

$\frac{\frac{1}{\sqrt{3}}(1)+1}{\frac{1}{\sqrt{3}}-1}$

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