If ` tan 5 theta = cot 3 theta, `then `theta` =

Options

A. `(2n + 1) pi/4`
B. `(2n + 1)pi/8`
C. `(2n + 1) pi/16`
D. none

Answer

To solve the equation \( \tan 5\theta = \cot 3\theta \), we can follow these steps:

### Step 1: Rewrite the cotangent in terms of tangent
We know that \( \cot x = \frac{1}{\tan x} \). Therefore, we can rewrite the equation as:
\[ \tan 5\theta = \frac{1}{\tan 3\theta} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ \tan 5\theta \cdot \tan 3\theta = 1 \]

### Step 3: Use the identity for tangent
Using the identity \( \tan A \tan B = 1 \) implies that \( A + B = \frac{\pi}{2} + n\pi \) for some integer \( n \). Thus, we can write:
\[ 5\theta + 3\theta = \frac{\pi}{2} + n\pi \]

### Step 4: Combine like terms
Combining the terms gives us:
\[ 8\theta = \frac{\pi}{2} + n\pi \]

### Step 5: Solve for \(\theta\)
Dividing both sides by 8, we have:
\[ \theta = \frac{\frac{\pi}{2} + n\pi}{8} \]
This simplifies to:
\[ \theta = \frac{\pi}{16} + \frac{n\pi}{8} \]

### Step 6: General solution
Thus, the general solution for \(\theta\) is:
\[ \theta = \frac{\pi}{16} + \frac{n\pi}{8}, \quad n \in \mathbb{Z} \]