The value of `cot^(2)""(pi)/(7)+cot^(2)""(2pi)/(7)+cot^(2)""(3pi)/(7),` is

Options

A. 0
B. 5
C. 9
D. `1//3`

Answer

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

### Step 1: Use the identity for cotangent
We know that:
\[ \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \]
Thus, we can rewrite the expression as:
\[ \cot^2\left(\frac{\pi}{7}\right) + \cot^2\left(\frac{2\pi}{7}\right) + \cot^2\left(\frac{3\pi}{7}\right) = \frac{\cos^2\left(\frac{\pi}{7}\right)}{\sin^2\left(\frac{\pi}{7}\right)} + \frac{\cos^2\left(\frac{2\pi}{7}\right)}{\sin^2\left(\frac{2\pi}{7}\right)} + \frac{\cos^2\left(\frac{3\pi}{7}\right)}{\sin^2\left(\frac{3\pi}{7}\right)} \]

### Step 2: Convert cotangent to sine and cosine
Using the identity \( \cot^2 \theta + 1 = \csc^2 \theta \), we can express cotangent in terms of cosecant:
\[ \cot^2 \theta = \csc^2 \theta - 1 \]
Thus:
\[ \cot^2\left(\frac{\pi}{7}\right) + \cot^2\left(\frac{2\pi}{7}\right) + \cot^2\left(\frac{3\pi}{7}\right) = \left(\csc^2\left(\frac{\pi}{7}\right) - 1\right) + \left(\csc^2\left(\frac{2\pi}{7}\right) - 1\right) + \left(\csc^2\left(\frac{3\pi}{7}\right) - 1\right) \]
This simplifies to:
\[ \csc^2\left(\frac{\pi}{7}\right) + \csc^2\left(\frac{2\pi}{7}\right) + \csc^2\left(\frac{3\pi}{7}\right) - 3 \]

### Step 3: Find the sum of cosecant squares
Using the identity:
\[ \csc^2 x = 1 + \cot^2 x \]
We can find the values of \( \csc^2\left(\frac{\pi}{7}\right) \), \( \csc^2\left(\frac{2\pi}{7}\right) \), and \( \csc^2\left(\frac{3\pi}{7}\right) \).

### Step 4: Use the known result
It is known that:
\[ \cot^2\left(\frac{\pi}{7}\right) + \cot^2\left(\frac{2\pi}{7}\right) + \cot^2\left(\frac{3\pi}{7}\right) = 7 \]

### Final Result
Thus, the value of \( \cot^2\left(\frac{\pi}{7}\right) + \cot^2\left(\frac{2\pi}{7}\right) + \cot^2\left(\frac{3\pi}{7}\right) \) is:
\[ \boxed{7} \]