The value of ` 2 cot^(2) (pi//6) + 4 tan^(2)(pi//6) -3 cosec (pi//6)` is

Options

A. 2
B. 4
C. `4//3`
D. `3//4`

Explanation

To solve the expression \( 2 \cot^2\left(\frac{\pi}{6}\right) + 4 \tan^2\left(\frac{\pi}{6}\right) - 3 \csc\left(\frac{\pi}{6}\right) \), we will first calculate the values of \( \cot\left(\frac{\pi}{6}\right) \), \( \tan\left(\frac{\pi}{6}\right) \), and \( \csc\left(\frac{\pi}{6}\right) \).

### Step 1: Calculate \( \cot\left(\frac{\pi}{6}\right) \)

\[ \cot\left(\frac{\pi}{6}\right) = \frac{1}{\tan\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

### Step 2: Calculate \( \tan\left(\frac{\pi}{6}\right) \)

\[ \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \]

### Step 3: Calculate \( \csc\left(\frac{\pi}{6}\right) \)

\[ \csc\left(\frac{\pi}{6}\right) = \frac{1}{\sin\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 4: Substitute the values into the expression

Now we substitute these values into the original expression:

\[ 2 \cot^2\left(\frac{\pi}{6}\right) + 4 \tan^2\left(\frac{\pi}{6}\right) - 3 \csc\left(\frac{\pi}{6}\right) \]

Substituting the values:

\[ = 2 (\sqrt{3})^2 + 4 \left(\frac{1}{\sqrt{3}}\right)^2 - 3(2) \]

### Step 5: Simplify the expression

Calculating each term:

\[ = 2 \cdot 3 + 4 \cdot \frac{1}{3} - 6 \]

\[ = 6 + \frac{4}{3} - 6 \]

### Step 6: Combine the terms

Now, we can combine the terms:

\[ = 0 + \frac{4}{3} = \frac{4}{3} \]

### Final Answer

Thus, the value of the expression is:

\[ \frac{4}{3} \]

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