If 15 ° + (1)/( 75 °) + (1)/( 105 °) + 195 ° = 2a , then the value...

Given:

$2a = \tan 15 ° + \dfrac{1}{\tan 75 °} + \dfrac{1}{\tan 105 °} + \tan 195 °$

$⇒ 2a = \tan 15 ° + \dfrac{1}{\tan \left(90 ° - 15 °\right)} + \dfrac{1}{\tan \left(90 ° + 15 °\right)} + \tan \left(180 ° + 15 °\right)$

$⇒ 2a = \tan 15 ° + \dfrac{1}{\cot 15 °} - \dfrac{1}{\cot 15 °} + \tan 15 °$

$⇒ 2a = 2\tan 15 °$

$⇒ a = \tan 15 °$

$⇒ a = \tan \left(45 ° - 30 °\right)$

$⇒ a = \dfrac{1 - \dfrac{1}{\sqrt{3}}}{1 + \dfrac{1}{\sqrt{3}}}$

$⇒ a = \dfrac{\sqrt{3} - 1}{\sqrt{3} + 1}$

So,

$a + \dfrac{1}{a} = \dfrac{\sqrt{3} - 1}{\sqrt{3} + 1} + \dfrac{\sqrt{3} + 1}{\sqrt{3} - 1}$

$⇒ a + \dfrac{1}{a} = \dfrac{{\left(\sqrt{3} - 1\right)}^{2} + {\left(\sqrt{3} + 1\right)}^{2}}{2}$

$⇒ a + \dfrac{1}{a} = \dfrac{8}{2} = 4$