Value of `2 sin ^(2) "" (pi)/(6) + cosec ^(2) "" (7pi)/(6) .cos^(2) "" (pi)/(3) is (m)/(m-1).` The value of 'm' is

Options

A. 3
B. 2
C. 4
D. None of these

Answer

To solve the problem, we need to find the value of the expression \(2 \sin^2 \left(\frac{\pi}{6}\right) + \csc^2 \left(\frac{7\pi}{6}\right) \cdot \cos^2 \left(\frac{\pi}{3}\right)\) and express it in the form \(\frac{m}{m-1}\) to find the value of \(m\).

### Step 1: Calculate \( \sin^2 \left(\frac{\pi}{6}\right) \)
We know that:
\[ \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Thus,
\[ \sin^2 \left(\frac{\pi}{6}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 2: Calculate \( 2 \sin^2 \left(\frac{\pi}{6}\right) \)
Now, substituting the value we found:
\[ 2 \sin^2 \left(\frac{\pi}{6}\right) = 2 \cdot \frac{1}{4} = \frac{1}{2} \]

### Step 3: Calculate \( \csc^2 \left(\frac{7\pi}{6}\right) \)
First, we need to find \(\sin \left(\frac{7\pi}{6}\right)\):
\[ \frac{7\pi}{6} = \pi + \frac{\pi}{6} \]
In the third quadrant, \(\sin\) is negative, so:
\[ \sin \left(\frac{7\pi}{6}\right) = -\sin \left(\frac{\pi}{6}\right) = -\frac{1}{2} \]
Thus,
\[ \csc \left(\frac{7\pi}{6}\right) = \frac{1}{\sin \left(\frac{7\pi}{6}\right)} = \frac{1}{-\frac{1}{2}} = -2 \]
Therefore,
\[ \csc^2 \left(\frac{7\pi}{6}\right) = (-2)^2 = 4 \]

### Step 4: Calculate \( \cos^2 \left(\frac{\pi}{3}\right) \)
We know that:
\[ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \]
Thus,
\[ \cos^2 \left(\frac{\pi}{3}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 5: Combine all parts of the expression
Now, we can substitute these values back into the original expression:
\[ 2 \sin^2 \left(\frac{\pi}{6}\right) + \csc^2 \left(\frac{7\pi}{6}\right) \cdot \cos^2 \left(\frac{\pi}{3}\right) = \frac{1}{2} + 4 \cdot \frac{1}{4} \]
Calculating the second term:
\[ 4 \cdot \frac{1}{4} = 1 \]
Thus, the expression simplifies to:
\[ \frac{1}{2} + 1 = \frac{1}{2} + \frac{2}{2} = \frac{3}{2} \]

### Step 6: Set the expression equal to \(\frac{m}{m-1}\)
We have:
\[ \frac{3}{2} = \frac{m}{m-1} \]

### Step 7: Cross-multiply to solve for \(m\)
Cross-multiplying gives:
\[ 3(m-1) = 2m \]
Expanding this, we have:
\[ 3m - 3 = 2m \]
Rearranging gives:
\[ 3m - 2m = 3 \implies m = 3 \]

### Final Answer
The value of \(m\) is:
\[ \boxed{3} \]