Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

Given:

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

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

$\Rightarrow 2a=\tan 15°+\frac{1}{\cot 15°}-\frac{1}{\cot 15°}+\tan 15°$

$\Rightarrow 2a=2\tan 15°$

$\Rightarrow a=\tan 15°$

$\Rightarrow a=\tan \left(45°-30°\right)$

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

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

So,

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

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

$\Rightarrow a+\frac{1}{a}=\frac{8}{2}=4$