Question
Find the value of $\tan 105^{\circ}$.
✓
Answer
$\begin{array}{l}\tan 105^{\circ}=\tan \left(60^{\circ}+45^{\circ}\right)=\frac{\tan 60^{\circ}+\tan 45^{\circ}}{1-\tan 60^{\circ} \tan 45^{\circ}} \\ \begin{aligned} \tan 105^{\circ} & =\frac{\sqrt{3}+1}{1-\sqrt{3} \times 1}=\frac{\sqrt{3}+1}{1-\sqrt{3}} \\ \tan 105^{\circ} & =\frac{\sqrt{3}+1}{1-\sqrt{3}} \times \frac{\sqrt{3}+1}{\sqrt{3}+1}=\frac{(\sqrt{3}+1)^2}{1-3} \\ & =\frac{(\sqrt{3})^2+2 \times \sqrt{3}+(1)^2}{-2} \\ & =\frac{3+2 \sqrt{3}+1}{-2}=\frac{4+2 \sqrt{3}}{-2} \\ & =\frac{2(2+\sqrt{3})}{-2}=-(2+\sqrt{3})\end{aligned}\end{array}$
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