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Q)

# Find the value of $\tan 105^{\large\circ}$

$\begin{array}{1 1}(A)\;2+\sqrt 3&(B)\;2-\sqrt 3\\(C)\;-(2+\sqrt 3)&(D)\;0\end{array}$

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A)
Toolbox:
• $\tan (A+B)=\large\frac{\tan A+\tan B}{1+\tan A\tan B}$
$\tan 105^{\large\circ}=\tan(60^{\large\circ}+45^{\large\circ})$
$\Rightarrow \large\frac{\tan 60^{\large\circ}+\tan 45^{\large\circ}}{1-\tan 60^{\large\circ}\tan 45^{\large\circ}}$
$\Rightarrow \large\frac{\sqrt 3+1}{1-\sqrt 3.1}$
$\Rightarrow \large\frac{\sqrt 3+1}{1-\sqrt 3}$
$\Rightarrow \large\frac{\sqrt 3+1}{1-\sqrt 3}\times \large\frac{1+\sqrt 3}{1+\sqrt 3}$
$\Rightarrow \large\frac{(\sqrt 3+1)^2}{1^2-(\sqrt 3)^2}$
$\Rightarrow \large\frac{ 3+1+2\sqrt 3}{1- 3}$
$\Rightarrow \large\frac{ 4+2\sqrt 3}{- 2}$
$\Rightarrow -(2+\sqrt 3)$
Hence (C) is the correct answer.