The value of $\tan 15 ° + \tan 75 ° + c o t 15 ° + c o t 75 °$

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answer is C.

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Detailed Solution

$\begin{array}{rcl} \tan 15 ° + c o t 75 ° + \tan 75 ° + c o t 15 ° & = & \tan 15 ° + \tan 15 ° + \tan 75 ° + \tan 75 ° \\ = & 2 (\tan 15 ° + \tan 75 °) \\ = & 2 (2 - \sqrt{2} + 2 + \sqrt{2}) \\ = & 2 (4) \\ = & 8 \end{array}$

$\begin{array}{rcl} \tan 15 ° + c o t 75 ° + \tan 75 ° + c o t 15 ° & = & \tan 15 ° + \tan 15 ° + \tan 75 ° + \tan 75 ° \\ = & 2 (\tan 15 ° + \tan 75 °) \\ = & 2 (\tan (45 ° - 30 °) + \tan (45 ° + 30 °)) \\ = & 2 (\frac{\tan 45 ° - \tan 30 °}{1 + \tan 45 ° \tan 30 °} + \frac{\tan 45 ° + \tan 30 °}{1 - \tan 45 ° \tan 30 °}) \\ = & 2 (\frac{1 - \tan 30 °}{1 + \tan 30 °} + \frac{1 + \tan 30 °}{1 - \tan 30 °}) \\ = & 2 (\frac{1 - \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}} + \frac{1 - \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}}) \\ = & 2 (\frac{\sqrt{3} - 1}{\sqrt{3} + 1} + \frac{\sqrt{3} + 1}{\sqrt{3} - 1}) \\ = & 2 (\frac{{(\sqrt{3} - 1)}^{2} + {(\sqrt{3} + 1)}^{2}}{3 - 1}) \\ = & 2 (3 + 1) \\ = & 8 \end{array}$