If `3 cot theta = 4 cos theta`, then what is the value of `cos 2theta`?

Options

A. `2//16`
B. `-1//8`
C. `7//16`
D. `9//16`

Answer

To solve the equation \(3 \cot \theta = 4 \cos \theta\) and find the value of \(\cos 2\theta\), we can follow these steps:

### Step 1: Rewrite the cotangent function
We know that:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Substituting this into the equation gives us:
\[ 3 \frac{\cos \theta}{\sin \theta} = 4 \cos \theta \]

### Step 2: Simplify the equation
Assuming \(\cos \theta \neq 0\), we can divide both sides by \(\cos \theta\):
\[ 3 \frac{1}{\sin \theta} = 4 \]
This simplifies to:
\[ \frac{3}{\sin \theta} = 4 \]

### Step 3: Solve for \(\sin \theta\)
Cross-multiplying gives:
\[ 3 = 4 \sin \theta \]
Thus, we can find \(\sin \theta\):
\[ \sin \theta = \frac{3}{4} \]

### Step 4: Use the Pythagorean identity to find \(\cos \theta\)
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ \left(\frac{3}{4}\right)^2 + \cos^2 \theta = 1 \]
This leads to:
\[ \frac{9}{16} + \cos^2 \theta = 1 \]
Subtracting \(\frac{9}{16}\) from both sides gives:
\[ \cos^2 \theta = 1 - \frac{9}{16} = \frac{16}{16} - \frac{9}{16} = \frac{7}{16} \]

### Step 5: Find \(\cos 2\theta\)
We can use the double angle formula for cosine:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
Substituting \(\cos^2 \theta = \frac{7}{16}\):
\[ \cos 2\theta = 2 \cdot \frac{7}{16} - 1 = \frac{14}{16} - 1 = \frac{14}{16} - \frac{16}{16} = \frac{-2}{16} = -\frac{1}{8} \]

### Final Answer
Thus, the value of \(\cos 2\theta\) is:
\[ \cos 2\theta = -\frac{1}{8} \]