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

The Correct Option is A

We are given the equation:

\[ \tan 15^\circ + \frac{1}{\tan 75^\circ} + \tan 105^\circ + \tan 195^\circ = 2a. \]

Step 1: Simplify the Individual Terms

We know that:

• \(\tan 15^\circ = 2 - \sqrt{3}\),
• \(\frac{1}{\tan 75^\circ} = \cot 75^\circ = 2 - \sqrt{3}\),
• \(\tan 105^\circ = \cot 75^\circ = -\cot 75^\circ = -2 + \sqrt{3}\),
• \(\tan 195^\circ = \tan(180^\circ + 15^\circ) = \tan 15^\circ = 2 - \sqrt{3}\).

Step 2: Substitute into the Equation

Substituting these values into the equation:

\[ (2 - \sqrt{3}) + (2 - \sqrt{3}) + (-2 + \sqrt{3}) + (2 - \sqrt{3}) = 2a. \]

Simplify the left-hand side:

\[ 2 + 2 - 2 + 2 - \sqrt{3} - \sqrt{3} + \sqrt{3} - \sqrt{3} = 2a, \]

which simplifies further to:

\[ 4 - 2\sqrt{3} = 2a. \]

Step 3: Solve for \(a\)

Dividing both sides by 2:

\[ a = 2 - \sqrt{3}. \]

Step 4: Calculate \(a + \frac{1}{a}\)

Now we calculate:

\[ a + \frac{1}{a} = (2 - \sqrt{3}) + \frac{1}{2 - \sqrt{3}}. \]

To simplify \(\frac{1}{2 - \sqrt{3}}\), rationalize the denominator:

\[ \frac{1}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = \frac{2 + \sqrt{3}}{(2 - \sqrt{3})(2 + \sqrt{3})}. \]

The denominator simplifies as follows:

\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 4 - 3 = 1. \]

Thus:

\[ \frac{1}{2 - \sqrt{3}} = 2 + \sqrt{3}. \]

Step 5: Combine the Results

Now substitute back into the expression for \(a + \frac{1}{a}\):

\[ a + \frac{1}{a} = (2 - \sqrt{3}) + (2 + \sqrt{3}) = 4. \]

Therefore, the value of \(a + \frac{1}{a}\) is 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