Find the value of tan 15^o - Vidyadip

Question

Find the value of tan $15^o$

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Answer

Let y = $\tan 15^{\circ}$, then
$y=\tan \left(45^{\circ}-30^{\circ}\right)$
$\tan (x-y)=\frac{\tan x-\tan y}{1+\tan x \tan y}$
$\tan 15^{\circ}=\tan \left(45^{\circ}-30^{\circ}\right)$
$\tan 15^{\circ}=\frac{\tan 45^{\circ}-\tan 30^{\circ}}{1+\tan 45^{\circ} \tan 30^{\circ}}$
$\tan 15^{\circ}=\frac{1-\frac{1}{\sqrt{3}}}{1+1 \times \frac{1}{\sqrt{3}}}=\frac{\sqrt{3}-1}{\sqrt{3}+1}$
$\begin{equation} \tan 15^{\circ}=\frac{(\sqrt{3}-1) \times(\sqrt{3}-1)}{(\sqrt{3}+1) \times(\sqrt{3}-1)}=\frac{(\sqrt{3}-1)^{2}}{(\sqrt{3})^{2}-1^{2}}=\frac{3+1-2 \sqrt{3}}{3-1}=\frac{2(2-\sqrt{3})}{2} \end{equation}$
$\begin{equation} \tan 15^{\circ}=2-\sqrt{3} \end{equation}$

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