If tan 15 ° + 1 tan 75 ° + 1 tan 105 ° + tan 195 ° = 2 a , then the value of ( a + 1 a ) is [2023]

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Solution

Q.
If $\tan 15 ° + \frac{1}{\tan 75 °} + \frac{1}{\tan 105 °} + \tan 195 ° = 2 a$ , then the value of $(a + \frac{1}{a})$ is [2023]

1 $4$

2 $2$

3 $4 - 2 \sqrt{3}$

4 $5 - \frac{3}{2} \sqrt{3}$

Ans.
(1)

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

We know that $\tan 15 ° = 2 - \sqrt{3}$

$\frac{1}{\tan 75 °} = \tan 15 ° = 2 - \sqrt{3},$ $\frac{1}{\tan 105 °} = - \cot 75 ° = \sqrt{3} - 2$

and $\tan 195 ° = \tan 15 ° = 2 - \sqrt{3}$

So, $2 (2 - \sqrt{3}) = 2 a \Rightarrow a = 2 - \sqrt{3}$

$\Rightarrow a + \frac{1}{a} = 2 - \sqrt{3} + \frac{1}{2 - \sqrt{3}} = 4$

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