Solve `cot theta + tan theta=2 cosec theta`.

Explanation

To solve the equation \( \cot \theta + \tan \theta = 2 \csc \theta \), we will follow these steps:

### Step 1: Rewrite the trigonometric functions
We start by rewriting \( \cot \theta \), \( \tan \theta \), and \( \csc \theta \) in terms of sine and cosine:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta} \]
Substituting these into the equation gives:
\[ \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} = 2 \cdot \frac{1}{\sin \theta} \]

### Step 2: Find a common denominator
The left-hand side has two fractions, so we find a common denominator:
\[ \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} = \frac{2}{\sin \theta} \]

### Step 3: Use the Pythagorean identity
We know from the Pythagorean identity that \( \sin^2 \theta + \cos^2 \theta = 1 \). Therefore, we can simplify the equation:
\[ \frac{1}{\sin \theta \cos \theta} = \frac{2}{\sin \theta} \]

### Step 4: Cross-multiply
Cross-multiplying gives:
\[ 1 = 2 \cos \theta \]

### Step 5: Solve for \( \cos \theta \)
Rearranging the equation, we find:
\[ \cos \theta = \frac{1}{2} \]

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

### Final Solution
Thus, the complete solution for \( \theta \) is:
\[ \theta = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad \theta = -\frac{\pi}{3} + 2n\pi, \quad n \in \mathbb{Z} \]