`cos(pi/14)+cos((3pi)/14)+cos((5pi)/14)=kcot(pi/14)` then `k` is equal to

Options

A. `1`
B. `1/2`
C. `2`
D. `-2`

Answer

To solve the equation \( \cos\left(\frac{\pi}{14}\right) + \cos\left(\frac{3\pi}{14}\right) + \cos\left(\frac{5\pi}{14}\right) = k \cot\left(\frac{\pi}{14}\right) \), we will follow these steps:

### Step 1: Write the expression
We start with the left-hand side:
\[ \cos\left(\frac{\pi}{14}\right) + \cos\left(\frac{3\pi}{14}\right) + \cos\left(\frac{5\pi}{14}\right) \]

### Step 2: Multiply and divide by \( 2 \sin\left(\frac{\pi}{14}\right) \)
To convert the expression into a form involving cotangent, we multiply and divide by \( 2 \sin\left(\frac{\pi}{14}\right) \):
\[ = \frac{2 \sin\left(\frac{\pi}{14}\right) \left(\cos\left(\frac{\pi}{14}\right) + \cos\left(\frac{3\pi}{14}\right) + \cos\left(\frac{5\pi}{14}\right)\right)}{2 \sin\left(\frac{\pi}{14}\right)} \]

### Step 3: Use the identity \( 2 \sin A \cos B = \sin(2A) \)
Now, we can use the identity \( 2 \sin A \cos B = \sin(2A) \):
\[ = \frac{1}{2 \sin\left(\frac{\pi}{14}\right)} \left( \sin\left(\frac{2\pi}{14}\right) + \sin\left(\frac{4\pi}{14}\right) + \sin\left(\frac{6\pi}{14}\right) \right) \]
This simplifies to:
\[ = \frac{1}{2 \sin\left(\frac{\pi}{14}\right)} \left( \sin\left(\frac{\pi}{7}\right) + \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{3\pi}{7}\right) \right) \]

### Step 4: Use the sum of sines identity
We know from trigonometric identities that:
\[ \sin A + \sin B + \sin C = 4 \sin\left(\frac{A+B+C}{2}\right) \sin\left(\frac{A-B}{2}\right) \sin\left(\frac{B-C}{2}\right) \]
Applying this to our angles:
\[ \sin\left(\frac{\pi}{7}\right) + \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{3\pi}{7}\right) = 4 \sin\left(\frac{6\pi}{21}\right) \sin\left(\frac{-\pi/7}{2}\right) \sin\left(\frac{-\pi/7}{2}\right) \]
This results in:
\[ = 4 \sin\left(\frac{2\pi}{7}\right) \sin\left(\frac{\pi}{14}\right) \]

### Step 5: Substitute back into the expression
Substituting back, we have:
\[ = \frac{1}{2 \sin\left(\frac{\pi}{14}\right)} \cdot 4 \sin\left(\frac{2\pi}{7}\right) \sin\left(\frac{\pi}{14}\right) \]
This simplifies to:
\[ = 2 \sin\left(\frac{2\pi}{7}\right) \]

### Step 6: Relate to cotangent
Now, we know that:
\[ \sin\left(\frac{2\pi}{7}\right) = \cot\left(\frac{\pi}{14}\right) \]
Thus, we can express:
\[ 2 \sin\left(\frac{2\pi}{7}\right) = k \cot\left(\frac{\pi}{14}\right) \]

### Step 7: Compare coefficients
From the equation, we can see that:
\[ k = 1 \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{1} \]