Question

If $\tan 15^{\circ }+\frac{1}{\tan 75^{\circ }}+\frac{1}{\tan 105^{\circ }}+\tan 195^{\circ }=2a$, 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)

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