The vlaue of `cosec^(2)""(pi)/(7)+cosec^(2)""(2pi)/(7)+cosec^(2)""(3pi)/(7),` is

Options

A. 20
B. 2
C. 22
D. 23

Answer

To find the value of \( \csc^2\left(\frac{\pi}{7}\right) + \csc^2\left(\frac{2\pi}{7}\right) + \csc^2\left(\frac{3\pi}{7}\right) \), we can follow these steps:

### Step 1: Rewrite Cosecant in Terms of Sine
We know that:
\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} \]
Thus, we can rewrite the expression as:
\[ \csc^2\left(\frac{\pi}{7}\right) + \csc^2\left(\frac{2\pi}{7}\right) + \csc^2\left(\frac{3\pi}{7}\right) = \frac{1}{\sin^2\left(\frac{\pi}{7}\right)} + \frac{1}{\sin^2\left(\frac{2\pi}{7}\right)} + \frac{1}{\sin^2\left(\frac{3\pi}{7}\right)} \]

### Step 2: Use the Identity for Sine
We can use the identity:
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]
So, we can rewrite each sine term:
\[ \sin^2\left(\frac{\pi}{7}\right) = \frac{1 - \cos\left(\frac{2\pi}{7}\right)}{2}, \quad \sin^2\left(\frac{2\pi}{7}\right) = \frac{1 - \cos\left(\frac{4\pi}{7}\right)}{2}, \quad \sin^2\left(\frac{3\pi}{7}\right) = \frac{1 - \cos\left(\frac{6\pi}{7}\right)}{2} \]

### Step 3: Substitute Back into the Expression
Now substituting back, we have:
\[ \frac{1}{\sin^2\left(\frac{\pi}{7}\right)} = \frac{2}{1 - \cos\left(\frac{2\pi}{7}\right)}, \quad \frac{1}{\sin^2\left(\frac{2\pi}{7}\right)} = \frac{2}{1 - \cos\left(\frac{4\pi}{7}\right)}, \quad \frac{1}{\sin^2\left(\frac{3\pi}{7}\right)} = \frac{2}{1 - \cos\left(\frac{6\pi}{7}\right)} \]

### Step 4: Combine the Terms
Thus, the expression becomes:
\[ \csc^2\left(\frac{\pi}{7}\right) + \csc^2\left(\frac{2\pi}{7}\right) + \csc^2\left(\frac{3\pi}{7}\right) = 2 \left( \frac{1}{1 - \cos\left(\frac{2\pi}{7}\right)} + \frac{1}{1 - \cos\left(\frac{4\pi}{7}\right)} + \frac{1}{1 - \cos\left(\frac{6\pi}{7}\right)} \right) \]

### Step 5: Use the Sum of Cosines
Using the identity for the sum of cosines, we can simplify the expression further. The sum of cosines can be expressed as:
\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = -\frac{1}{2} \]

### Step 6: Final Calculation
Now we can substitute these values into the expression and simplify to find the final result. After performing the calculations, we find that:
\[ \csc^2\left(\frac{\pi}{7}\right) + \csc^2\left(\frac{2\pi}{7}\right) + \csc^2\left(\frac{3\pi}{7}\right) = 8 \]

Thus, the final answer is:
\[ \boxed{8} \]