 # What is the value of the following expression:sqrt3cot15^circ Nann 2021-09-13 Answered
What is the value of the following expression:
$\sqrt{3}{\mathrm{cot}15}^{\circ }$
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First, find ${\mathrm{tan}15}^{\circ }$
$\mathrm{tan}\left(A-B\right)=\frac{\mathrm{tan}A-\mathrm{tan}B}{1+\mathrm{tan}A\mathrm{tan}B}$
${\mathrm{tan}15}^{\circ }=\mathrm{tan}\left({60}^{\circ }-{45}^{\circ }\right)$
${\mathrm{tan}15}^{\circ }=\frac{{\mathrm{tan}60}^{\circ }-{\mathrm{tan}45}^{\circ }}{1+{\mathrm{tan}60}^{\circ }{\mathrm{tan}45}^{\circ }}$
${\mathrm{tan}15}^{\circ }=\frac{\sqrt{3}-1}{1+\left(\sqrt{3}\right)\left(1\right)}$
$=\frac{\sqrt{3}-1}{\sqrt{3}+1}$
Then,
${\mathrm{cot}15}^{\circ }=\frac{\sqrt{3}+1}{\sqrt{3}-1}$
${\mathrm{cot}15}^{\circ }=\frac{\sqrt{3}+1}{\sqrt{3}-1}×\frac{\sqrt{3}+1}{\sqrt{3}+1}$
${\mathrm{cot}15}^{\circ }=\frac{3+2\sqrt{3}+1}{3-1}$
${\mathrm{cot}15}^{\circ }=\frac{2\sqrt{3}+4}{2}$
$c\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}{15}^{\circ }=\sqrt{3}+2$
Substitute to given expression:
$\sqrt{3}{\mathrm{cot}15}^{\circ }=\sqrt{3}\left(\sqrt{3}+2\right)$
$\sqrt{3}{\mathrm{cot}15}^{\circ }=3+2\sqrt{2}$