Solve the general vlaue.
`cot theta + tan theta = 2 cosec theta `

Answer

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

### Step 1: Rewrite the trigonometric functions in terms of sine and cosine.
We know that:
- \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
- \( \csc \theta = \frac{1}{\sin \theta} \)

Substituting these into the equation gives us:
\[ \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} = 2 \cdot \frac{1}{\sin \theta} \]

### Step 2: Find a common denominator for the left-hand side.
The common denominator for \( \sin \theta \) and \( \cos \theta \) is \( \sin \theta \cos \theta \). Thus, we can rewrite the left-hand side:
\[ \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta} \]
(using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \)).

### Step 3: Cross-multiply to eliminate the fractions.
Cross-multiplying gives us:
\[ \cos^2 \theta + \sin^2 \theta = \sin \theta \cos \theta \]
This simplifies to:
\[ 1 = \sin \theta \cos \theta \]

### Step 4: Use the double angle identity.
We can use the identity \( \sin(2\theta) = 2 \sin \theta \cos \theta \) to rewrite the equation:
\[ 1 = \frac{1}{2} \sin(2\theta) \]
Multiplying both sides by 2 gives:
\[ \sin(2\theta) = 2 \]

### Step 5: Analyze the equation.
The sine function can only take values between -1 and 1. Therefore, there are no solutions for \( \sin(2\theta) = 2 \).

### Conclusion:
Since \( \sin(2\theta) = 2 \) has no solutions, the original equation \( \cot \theta + \tan \theta = 2 \csc \theta \) has no solutions.