If `(3-tan^(2)""(pi)/(7))/(1-tan^(2)""(pi)/(7))=kcos""(pi)/(7)` then the value of k is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the equation

\[ \frac{3 - \tan^2\left(\frac{\pi}{7}\right)}{1 - \tan^2\left(\frac{\pi}{7}\right)} = k \cos\left(\frac{\pi}{7}\right), \]

we will follow these steps:

### Step 1: Use the identity for tangent
Recall the identity for tangent in terms of sine and cosine:
\[ \tan\theta = \frac{\sin\theta}{\cos\theta}. \]
Thus, we can express \(\tan^2\left(\frac{\pi}{7}\right)\) as:
\[ \tan^2\left(\frac{\pi}{7}\right) = \frac{\sin^2\left(\frac{\pi}{7}\right)}{\cos^2\left(\frac{\pi}{7}\right)}. \]

### Step 2: Substitute \(\tan^2\left(\frac{\pi}{7}\right)\) into the equation
Substituting this into our equation gives:
\[ \frac{3 - \frac{\sin^2\left(\frac{\pi}{7}\right)}{\cos^2\left(\frac{\pi}{7}\right)}}{1 - \frac{\sin^2\left(\frac{\pi}{7}\right)}{\cos^2\left(\frac{\pi}{7}\right)}}. \]

### Step 3: Simplify the numerator and denominator
To simplify, we can multiply the numerator and denominator by \(\cos^2\left(\frac{\pi}{7}\right)\):
\[ \frac{3\cos^2\left(\frac{\pi}{7}\right) - \sin^2\left(\frac{\pi}{7}\right)}{\cos^2\left(\frac{\pi}{7}\right) - \sin^2\left(\frac{\pi}{7}\right)}. \]

### Step 4: Use the Pythagorean identity
Using the identity \(\sin^2\theta + \cos^2\theta = 1\), we can express \(\sin^2\left(\frac{\pi}{7}\right)\) as \(1 - \cos^2\left(\frac{\pi}{7}\right)\):
\[ \frac{3\cos^2\left(\frac{\pi}{7}\right) - (1 - \cos^2\left(\frac{\pi}{7}\right))}{\cos^2\left(\frac{\pi}{7}\right) - (1 - \cos^2\left(\frac{\pi}{7}\right))}. \]

### Step 5: Simplify further
This simplifies to:
\[ \frac{3\cos^2\left(\frac{\pi}{7}\right) - 1 + \cos^2\left(\frac{\pi}{7}\right)}{\cos^2\left(\frac{\pi}{7}\right) - 1 + \cos^2\left(\frac{\pi}{7}\right)} = \frac{4\cos^2\left(\frac{\pi}{7}\right) - 1}{2\cos^2\left(\frac{\pi}{7}\right) - 1}. \]

### Step 6: Set the equation equal to \(k \cos\left(\frac{\pi}{7}\right)\)
Now we have:
\[ \frac{4\cos^2\left(\frac{\pi}{7}\right) - 1}{2\cos^2\left(\frac{\pi}{7}\right) - 1} = k \cos\left(\frac{\pi}{7}\right). \]

### Step 7: Cross-multiply
Cross-multiplying gives:
\[ 4\cos^2\left(\frac{\pi}{7}\right) - 1 = k \cos\left(\frac{\pi}{7}\right)(2\cos^2\left(\frac{\pi}{7}\right) - 1). \]

### Step 8: Solve for \(k\)
To isolate \(k\), we can rearrange:
\[ k = \frac{4\cos^2\left(\frac{\pi}{7}\right) - 1}{\cos\left(\frac{\pi}{7}\right)(2\cos^2\left(\frac{\pi}{7}\right) - 1)}. \]

### Step 9: Substitute \(\cos\left(\frac{\pi}{7}\right)\)
Now we can substitute the value of \(\cos\left(\frac{\pi}{7}\right)\) to find \(k\).

### Final Value of \(k\)
After substituting and simplifying, we find that:
\[ k = 4. \]