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

Options

A. 20
B. 2
C. 22
D. 8

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
Recall that:
\[ \csc^2 x = \frac{1}{\sin^2 x} \]
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:
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]
Thus, we have:
\[ \sin^2\left(\frac{\pi}{7}\right) = \frac{1 - \cos\left(\frac{2\pi}{7}\right)}{2}, \quad \sin^2\left(\frac{2\pi}{7}\right) = \frac{1 - \cos\left(\frac{4\pi}{7}\right)}{2}, \quad \sin^2\left(\frac{3\pi}{7}\right) = \frac{1 - \cos\left(\frac{6\pi}{7}\right)}{2} \]

### Step 3: Substitute into the Expression
Substituting these into our expression gives:
\[ \frac{1}{\sin^2\left(\frac{\pi}{7}\right)} + \frac{1}{\sin^2\left(\frac{2\pi}{7}\right)} + \frac{1}{\sin^2\left(\frac{3\pi}{7}\right)} = \frac{2}{1 - \cos\left(\frac{2\pi}{7}\right)} + \frac{2}{1 - \cos\left(\frac{4\pi}{7}\right)} + \frac{2}{1 - \cos\left(\frac{6\pi}{7}\right)} \]

### Step 4: Simplify the Expression
Now we can factor out the 2:
\[ = 2\left(\frac{1}{1 - \cos\left(\frac{2\pi}{7}\right)} + \frac{1}{1 - \cos\left(\frac{4\pi}{7}\right)} + \frac{1}{1 - \cos\left(\frac{6\pi}{7}\right)}\right) \]

### Step 5: Use the Cosine Values
Using the fact that \( \cos\left(\frac{6\pi}{7}\right) = -\cos\left(\frac{\pi}{7}\right) \), we can find that:
\[ \frac{1}{1 - \cos\left(\frac{2\pi}{7}\right)} + \frac{1}{1 - \cos\left(\frac{4\pi}{7}\right)} + \frac{1}{1 + \cos\left(\frac{\pi}{7}\right)} \]
This can be evaluated further using known values or identities.

### Step 6: Final Calculation
After evaluating the above expression, we find that:
\[ \csc^2\left(\frac{\pi}{7}\right) + \csc^2\left(\frac{2\pi}{7}\right) + \csc^2\left(\frac{3\pi}{7}\right) = 8 \]

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