Question

If $\tan 15^{\circ}+\frac{1}{\tan 75^{\circ}}+\frac{1}{\tan 105^{\circ}}+\tan 195^{\circ}=2 a$, then the value of $\left(a+\frac{1}{a}\right)$ is :

• A. 4
• B. 2
• C. $4-2 \sqrt{3}$
• D. $5-\frac{3}{2} \sqrt{3}$

Answer 1

Answer: (A)

$\tan 15^{\circ}=2-\sqrt{3}$
$ \frac{1}{\tan 75^{\circ}}=\cot 75^{\circ}=2-\sqrt{3} $
$ \frac{1}{\tan 105^{\circ}}=\cot \left(105^{\circ}\right)=-\cot 75^{\circ}=\sqrt{3}-2$
$ \tan 195^{\circ}=\tan 15^{\circ}=2-\sqrt{3}$
$ \therefore 2(2-\sqrt{3})=2 a \Rightarrow a=2-\sqrt{3} $
$\Rightarrow a+\frac{1}{a}=4$