General solution of the equation
`4 cot 2 theta = cot^(2) theta - tan^(2) theta` is ` theta` =

Options

A. `npi +- (pi//2)`
B. `npi +- (pi//3)`
C. `npi +- (pi//4)`
D. none of these

Answer

To solve the equation \( 4 \cot(2\theta) = \cot^2(\theta) - \tan^2(\theta) \), we will follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that:
\[ \cot(2\theta) = \frac{\cos(2\theta)}{\sin(2\theta)} = \frac{1 - \tan^2(\theta)}{2\tan(\theta)} \]
Thus, we can rewrite the left side:
\[ 4 \cot(2\theta) = 4 \cdot \frac{1 - \tan^2(\theta)}{2\tan(\theta)} = \frac{4(1 - \tan^2(\theta))}{2\tan(\theta)} = \frac{2(1 - \tan^2(\theta))}{\tan(\theta)} \]

### Step 2: Rewrite the right side
The right side of the equation is:
\[ \cot^2(\theta) - \tan^2(\theta) = \frac{\cos^2(\theta)}{\sin^2(\theta)} - \frac{\sin^2(\theta)}{\cos^2(\theta)} \]
Finding a common denominator gives:
\[ \frac{\cos^4(\theta) - \sin^4(\theta)}{\sin^2(\theta)\cos^2(\theta)} \]
Using the difference of squares:
\[ \cos^4(\theta) - \sin^4(\theta) = (\cos^2(\theta) - \sin^2(\theta))(\cos^2(\theta) + \sin^2(\theta)) = (\cos^2(\theta) - \sin^2(\theta)) \]
since \( \cos^2(\theta) + \sin^2(\theta) = 1 \).

### Step 3: Set the two sides equal
Now we equate both sides:
\[ \frac{2(1 - \tan^2(\theta))}{\tan(\theta)} = \frac{\cos^2(\theta) - \sin^2(\theta)}{\sin^2(\theta)\cos^2(\theta)} \]

### Step 4: Cross-multiply
Cross-multiplying gives:
\[ 2(1 - \tan^2(\theta)) \cdot \sin^2(\theta) \cos^2(\theta) = \tan(\theta)(\cos^2(\theta) - \sin^2(\theta)) \]

### Step 5: Substitute \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
Substituting \(\tan(\theta)\) into the equation:
\[ 2(1 - \frac{\sin^2(\theta)}{\cos^2(\theta)}) \cdot \sin^2(\theta) \cos^2(\theta) = \frac{\sin(\theta)}{\cos(\theta)}(\cos^2(\theta) - \sin^2(\theta)) \]

### Step 6: Simplify the equation
This can be simplified to:
\[ 2\sin^2(\theta) \cos^2(\theta) - 2\sin^4(\theta) = \sin(\theta)(\cos^2(\theta) - \sin^2(\theta)) \]
Rearranging gives us a polynomial equation in terms of \(\sin(\theta)\) and \(\cos(\theta)\).

### Step 7: Solve for \(\tan(\theta)\)
From the equation, we can find:
\[ \tan^2(\theta) = 1 \]
Thus, \(\tan(\theta) = 1\) or \(\tan(\theta) = -1\).

### Step 8: Find the general solution
The general solutions for \(\tan(\theta) = 1\) and \(\tan(\theta) = -1\) are:
\[ \theta = n\pi + \frac{\pi}{4} \quad \text{and} \quad \theta = n\pi - \frac{\pi}{4} \]
Combining these gives:
\[ \theta = n\pi \pm \frac{\pi}{4} \]

### Final Answer
Thus, the general solution is:
\[ \theta = n\pi \pm \frac{\pi}{4}, \quad n \in \mathbb{Z} \]