Find $\tan 75^{\large\circ}+\cot 75^{\large\circ}$

$(a)\;2\qquad(b)\;3\qquad(c)\;4\qquad(d)\;5$

$\tan 75^{\large\circ}=\tan(45^{\large\circ}+30^{\large\circ})$
$\qquad\quad=\large\frac{\tan 45^{\large\circ}+\tan 30^{\large\circ}}{1-\tan 45^{\large\circ}\tan 30^{\large\circ}}$
$\qquad\quad=\large\frac{1+\Large\frac{1}{\sqrt 3}}{1-\Large\frac{1}{\sqrt 3}}$
$\qquad\qquad=\large\frac{\sqrt 3+1}{\sqrt 3-1}$
$\cot 75^{\large\circ}=\large\frac{1}{\tan 75^{\Large\circ}}$
$\qquad\quad=\large\frac{\sqrt 3-1}{\sqrt 3+1}$
$\tan 75^{\large\circ}+\cot 75^{\large\circ}=\large\frac{\sqrt 3+1}{\sqrt 3-1}+\frac{\sqrt 3-1}{\sqrt 3+1}$
$\qquad\qquad\qquad=\large\frac{(\sqrt 3+1)^2+(\sqrt 3-1)^2}{(\sqrt 3-1)(\sqrt 3+1)}$
$\qquad\qquad\qquad=\large\frac{(4+2\sqrt 3)^2(4-2}{3-1}$
$\qquad\qquad\qquad=\large\frac{8}{2}$
$\qquad\qquad\qquad=4$
Hence (c) is the correct option.