Q182 / 270 · NDA 2012 (Phase 1) · Mathematics

What is $\tan \left(\frac{\pi}{12}\right)$ equal to?

  • Correct answer
    A
    $2-\sqrt{3}$
  • B
    $2+\sqrt{3}$
  • C
    $\sqrt{2}-\sqrt{3}$
  • D
    $\sqrt{3}-\sqrt{2}$

Solution

$\begin{aligned} & \tan \left(\frac{\pi}{3}-\frac{\pi}{4}\right) \\ & =\frac{\tan \frac{\pi}{3}-\tan \frac{\pi}{4}}{1+\tan \frac{\pi}{3} \tan \frac{\pi}{4}} \\ & =\frac{\sqrt{3}-1}{\sqrt{3}+1} \times \frac{\sqrt{3}-1}{\sqrt{3}-1} \\ & =\frac{(\sqrt{3}-1)^2}{3-1}=\frac{4-2 \sqrt{3}}{2}\end{aligned}$ $=\frac{2(2-\sqrt{3})}{2}=2-\sqrt{3}$.

Hence, the correct answer is option: 'A'.

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