If `3cottheta=4costheta`, what is the value of `cos2theta`?

Options

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

Answer

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

### Step 1: Rearranging the equation
We start with the equation:
\[ 3 \cos \theta = 4 \cos \theta \]
Subtract \(3 \cos \theta\) from both sides:
\[ 0 = 4 \cos \theta - 3 \cos \theta \]
This simplifies to:
\[ 0 = \cos \theta \]

### Step 2: Finding \(\sin \theta\)
Since \(\cos \theta = 0\), we know that \(\theta\) can be \(90^\circ\) or \(270^\circ\) (or \(\frac{\pi}{2}\) or \(\frac{3\pi}{2}\) in radians). For these angles:
\[ \sin \theta = 1 \quad \text{(for } \theta = 90^\circ\text{)} \]
\[ \sin \theta = -1 \quad \text{(for } \theta = 270^\circ\text{)} \]

### Step 3: Using the double angle formula
Now, we need to find \(\cos 2\theta\). We can use the double angle formula:
\[ \cos 2\theta = 1 - 2 \sin^2 \theta \]

### Step 4: Calculating \(\cos 2\theta\) for both angles
1. For \(\theta = 90^\circ\):
\[ \sin \theta = 1 \implies \cos 2\theta = 1 - 2(1^2) = 1 - 2 = -1 \]

2. For \(\theta = 270^\circ\):
\[ \sin \theta = -1 \implies \cos 2\theta = 1 - 2(-1)^2 = 1 - 2 = -1 \]

### Final Result
In both cases, we find that:
\[ \cos 2\theta = -1 \]