Trigonometric Ratios Question 3
Question 3 - 30 January - Shift 1
If $\tan 15^{\circ}+\frac{1}{\tan 75^{\circ}}+\frac{1}{\tan 105^{\circ}}+\tan 195^{\circ}=2 a$, then the value of $(a+\frac{1}{a})$ is :
(1) 4
(2) $4-2 \sqrt{3}$
(3) 2
(4) $5-\frac{3}{2} \sqrt{3}$
Show Answer
Answer: (1)
Solution:
Formula: Important Trigonometric Ratios, Periodic Formulas, Reciprocal Identities
Option (1)
$ \begin{aligned} & \tan 15^{\circ}=2-\sqrt{3} \\ & \frac{1}{\tan 75^{\circ}}=\cot 75^{\circ}=2-\sqrt{3} \\ & \frac{1}{\tan 105^{\circ}}=\cot (105^{\circ})=-\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 \end{aligned} $
