If `7theta=(2n+1)pi`, where `n=0,1,2,3,4,5,6`, then answer the following questions.
The value of `sec. (pi)/(7)+ sec. (3pi)/(7) + sec. (5pi)/(7) ` is

Options

A. (a)4
B. (b)-4
C. (c)3
D. (d)-3

Answer

To solve the problem, we need to find the value of \( \sec\left(\frac{\pi}{7}\right) + \sec\left(\frac{3\pi}{7}\right) + \sec\left(\frac{5\pi}{7}\right) \).

### Step-by-Step Solution:

1. **Understanding the Given Equation**:
We are given that \( 7\theta = (2n + 1)\pi \) for \( n = 0, 1, 2, 3, 4, 5, 6 \). This implies that:
\[ \theta = \frac{(2n + 1)\pi}{7} \]
This gives us the angles \( \theta \) for \( n = 0, 1, 2, 3, 4, 5, 6 \) as:
- \( \theta_0 = \frac{\pi}{7} \)
- \( \theta_1 = \frac{3\pi}{7} \)
- \( \theta_2 = \frac{5\pi}{7} \)
- \( \theta_3 = \frac{7\pi}{7} = \pi \)
- \( \theta_4 = \frac{9\pi}{7} \)
- \( \theta_5 = \frac{11\pi}{7} \)
- \( \theta_6 = \frac{13\pi}{7} \)

2. **Finding the Secant Values**:
We need to evaluate:
\[ \sec\left(\frac{\pi}{7}\right) + \sec\left(\frac{3\pi}{7}\right) + \sec\left(\frac{5\pi}{7}\right) \]

3. **Using a Polynomial Identity**:
We can use the identity involving the roots of a polynomial. The roots of the equation \( \cos(7\theta) = 0 \) are \( \theta = \frac{(2n + 1)\pi}{7} \). The corresponding secant values can be expressed in terms of a polynomial.

4. **Constructing the Polynomial**:
The polynomial whose roots are \( \sec\left(\frac{\pi}{7}\right), \sec\left(\frac{3\pi}{7}\right), \sec\left(\frac{5\pi}{7}\right) \) can be derived from:
\[ y^3 - 4y^2 - 4y + 8 = 0 \]
This polynomial can be derived from the properties of the angles involved.

5. **Finding the Sum of Secants**:
By Vieta's formulas, the sum of the roots (which are \( \sec\left(\frac{\pi}{7}\right), \sec\left(\frac{3\pi}{7}\right), \sec\left(\frac{5\pi}{7}\right) \)) is given by:
\[ \sec\left(\frac{\pi}{7}\right) + \sec\left(\frac{3\pi}{7}\right) + \sec\left(\frac{5\pi}{7}\right) = 4 \]

### Final Answer:
Thus, the value of \( \sec\left(\frac{\pi}{7}\right) + \sec\left(\frac{3\pi}{7}\right) + \sec\left(\frac{5\pi}{7}\right) \) is \( \boxed{4} \).