If `alpha=pi/7` then find the value of `(1/cosalpha+(2cosalpha)/(cos2alpha))`

Answer

To solve the problem, we need to find the value of the expression \( \frac{1}{\cos \alpha} + \frac{2 \cos \alpha}{\cos 2\alpha} \) where \( \alpha = \frac{\pi}{7} \).

### Step-by-Step Solution:

1. **Substitute the value of \(\alpha\)**:
\[ \alpha = \frac{\pi}{7} \]
Therefore, we need to calculate \( \cos \frac{\pi}{7} \) and \( \cos \frac{2\pi}{7} \).

2. **Use the double angle formula for cosine**:
The double angle formula states that:
\[ \cos 2\alpha = 2 \cos^2 \alpha - 1 \]
Thus, we can express \( \cos 2\alpha \) in terms of \( \cos \alpha \):
\[ \cos 2\alpha = 2 \cos^2 \left(\frac{\pi}{7}\right) - 1 \]

3. **Rewrite the expression**:
The expression can be rewritten as:
\[ \frac{1}{\cos \alpha} + \frac{2 \cos \alpha}{\cos 2\alpha} = \frac{1}{\cos \alpha} + \frac{2 \cos \alpha}{2 \cos^2 \alpha - 1} \]

4. **Find a common denominator**:
The common denominator for the two fractions is \( \cos \alpha (2 \cos^2 \alpha - 1) \):
\[ \frac{(2 \cos^2 \alpha - 1) + 2 \cos^2 \alpha}{\cos \alpha (2 \cos^2 \alpha - 1)} = \frac{4 \cos^2 \alpha - 1}{\cos \alpha (2 \cos^2 \alpha - 1)} \]

5. **Substitute the value of \( \cos \alpha \)**:
We can approximate \( \cos \frac{\pi}{7} \). For simplicity, let's denote \( \cos \frac{\pi}{7} \) as \( c \). Using a calculator or known values, we find:
\[ c \approx 0.90097 \]

6. **Calculate \( \cos 2\alpha \)**:
\[ \cos 2\alpha = 2c^2 - 1 \approx 2(0.90097^2) - 1 \approx 2(0.81174) - 1 \approx 0.62348 \]

7. **Substitute \( c \) into the expression**:
Now substituting \( c \) into the expression:
\[ \frac{4(0.90097^2) - 1}{0.90097(2(0.90097^2) - 1)} \approx \frac{4(0.81174) - 1}{0.90097(0.62348)} \]
Calculating the numerator:
\[ 4(0.81174) - 1 \approx 3.247 - 1 \approx 2.247 \]
And the denominator:
\[ 0.90097 \times 0.62348 \approx 0.560 \]

8. **Final calculation**:
Now, we compute:
\[ \frac{2.247}{0.560} \approx 4.01 \]

### Final Answer:
The value of \( \frac{1}{\cos \alpha} + \frac{2 \cos \alpha}{\cos 2\alpha} \) is approximately \( 4.01 \).