If: `(1-cos2x)/(1+cos 2x)=3,` then: x=

Options

A. `2npi +- pi/6`
B. `npi +- pi/6`
C. `2npi +- pi/3`
D. `npi +- pi/3`

Answer

To solve the equation \(\frac{1 - \cos 2x}{1 + \cos 2x} = 3\), we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ \frac{1 - \cos 2x}{1 + \cos 2x} = 3 \]

### Step 2: Cross-multiply
Cross-multiply to eliminate the fraction:
\[ 1 - \cos 2x = 3(1 + \cos 2x) \]

### Step 3: Expand the right side
Distribute the 3 on the right side:
\[ 1 - \cos 2x = 3 + 3\cos 2x \]

### Step 4: Rearrange the equation
Move all terms involving \(\cos 2x\) to one side and constant terms to the other:
\[ 1 - 3 = 3\cos 2x + \cos 2x \]
\[ -2 = 4\cos 2x \]

### Step 5: Solve for \(\cos 2x\)
Divide both sides by 4:
\[ \cos 2x = -\frac{1}{2} \]

### Step 6: Find the general solution for \(2x\)
The cosine function equals \(-\frac{1}{2}\) at specific angles:
\[ 2x = \frac{2\pi}{3} + 2n\pi \quad \text{and} \quad 2x = \frac{4\pi}{3} + 2n\pi \]
where \(n\) is any integer.

### Step 7: Solve for \(x\)
Divide each part of the equation by 2:
\[ x = \frac{\pi}{3} + n\pi \quad \text{and} \quad x = \frac{2\pi}{3} + n\pi \]

### Step 8: Combine the solutions
The general solutions can be written as:
\[ x = n\pi + \frac{\pi}{3} \quad \text{and} \quad x = n\pi + \frac{2\pi}{3} \]

### Final Answer
Thus, the solutions for \(x\) are:
\[ x = n\pi + \frac{\pi}{3} \quad \text{and} \quad x = n\pi + \frac{2\pi}{3} \]

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