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 1: Write the expression with a common denominator
We can combine the two fractions by finding a common denominator, which is \(\cos \alpha \cdot \cos 2\alpha\):

\[ \frac{1}{\cos \alpha} + \frac{2 \cos \alpha}{\cos 2\alpha} = \frac{\cos 2\alpha + 2 \cos^2 \alpha}{\cos \alpha \cdot \cos 2\alpha} \]

### Step 2: Use the double angle identity
We know that:

\[ \cos 2\alpha = 2 \cos^2 \alpha - 1 \]

Substituting this into our expression gives:

\[ \cos 2\alpha + 2 \cos^2 \alpha = (2 \cos^2 \alpha - 1) + 2 \cos^2 \alpha = 4 \cos^2 \alpha - 1 \]

### Step 3: Substitute back into the expression
Now we can substitute this back into our combined fraction:

\[ \frac{4 \cos^2 \alpha - 1}{\cos \alpha \cdot \cos 2\alpha} \]

### Step 4: Multiply numerator and denominator by \(2 \sin \alpha\)
To simplify further, we can multiply both the numerator and denominator by \(2 \sin \alpha\):

\[ \frac{2 \sin \alpha (4 \cos^2 \alpha - 1)}{2 \sin \alpha \cdot \cos \alpha \cdot \cos 2\alpha} \]

### Step 5: Recognize the sine double angle identity
The numerator can be simplified using the identity \(2 \sin \alpha \cos \alpha = \sin 2\alpha\):

\[ \frac{2 \sin \alpha (4 \cos^2 \alpha - 1)}{\sin 2\alpha \cdot \cos 2\alpha} \]

### Step 6: Use the sine addition formula
Using the identity \(2 \sin A \cos B = \sin(A + B) - \sin(A - B)\):

\[ \frac{4 \sin 3\alpha}{\sin 4\alpha} \]

### Step 7: Substitute \(\alpha = \frac{\pi}{7}\)
Now, substituting \(\alpha = \frac{\pi}{7}\):

\[ \frac{4 \sin \left(3 \cdot \frac{\pi}{7}\right)}{\sin \left(4 \cdot \frac{\pi}{7}\right)} \]

### Step 8: Simplify using sine properties
We know that:

\[ \sin \left( \pi - x \right) = \sin x \]

Thus:

\[ \sin \left( 4 \cdot \frac{\pi}{7} \right) = \sin \left( \pi - \frac{3\pi}{7} \right) = \sin \left( \frac{3\pi}{7} \right) \]

### Step 9: Final simplification
This leads to:

\[ \frac{4 \sin \left( \frac{3\pi}{7} \right)}{\sin \left( \frac{3\pi}{7} \right)} = 4 \]

### Conclusion
Thus, the value of the expression is:

\[ \boxed{4} \]