solve for `theta : tanptheta =cotqtheta`

Answer

To solve the equation \( \tan p\theta = \cot q\theta \), we will follow these steps:

### Step 1: Rewrite the cotangent
We know that \( \cot q\theta = \frac{1}{\tan q\theta} \). Therefore, we can rewrite the equation as:
\[ \tan p\theta = \frac{1}{\tan q\theta} \]

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

### Step 3: Rearrange the equation
We can rearrange this equation to:
\[ \tan p\theta \tan q\theta - 1 = 0 \]

### Step 4: Use the identity for tangent of a sum
Using the identity for the tangent of a sum, we have:
\[ \tan(p\theta + q\theta) = \frac{\tan p\theta + \tan q\theta}{1 - \tan p\theta \tan q\theta} \]
Since \( \tan p\theta \tan q\theta = 1 \), we can substitute this into our equation:
\[ \tan(p\theta + q\theta) = \frac{\tan p\theta + \tan q\theta}{1 - 1} = \frac{\tan p\theta + \tan q\theta}{0} \]

### Step 5: Determine when tangent is undefined
The tangent function is undefined when its denominator is zero. Therefore, we have:
\[ p\theta + q\theta = \frac{\pi}{2} + n\pi \]
where \( n \) is any integer.

### Step 6: Solve for \( \theta \)
Now, we can solve for \( \theta \):
\[ p\theta + q\theta = \frac{\pi}{2} + n\pi \]
\[ \theta(p + q) = \frac{\pi}{2} + n\pi \]
\[ \theta = \frac{\frac{\pi}{2} + n\pi}{p + q} \]

### Final Answer
Thus, the solution for \( \theta \) is:
\[ \theta = \frac{\frac{\pi}{2} + n\pi}{p + q}, \quad n \in \mathbb{Z} \]

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