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)

4

Answer 2

$tan15^∘=2−\sqrt3​$

$\frac1{tan75^∘}=cot75^∘=2−\sqrt3​$

$\frac1{tan105^∘}​=cot(105^∘)=−cot75^∘=\sqrt3​ −2$

$tan195^∘=tan15^∘=2−\sqrt3​$

$∴2(2−\sqrt3​ ​)=2a$

$⇒a=2−\sqrt3​ ​$

$⇒a+\frac1{a​}= \frac{2-\sqrt3​ }1 + \frac1{2-\sqrt3​ } =\frac{8-4\sqrt3​ }{2-\sqrt3​ }$

$=\frac{8-4\sqrt3​ }{2-\sqrt3​ } \times \frac{2+\sqrt3​ }{2+\sqrt3​ }= \frac{4}1$

$=$ $4$

Therefore, The correct answer is option (A) : 4