Solve the equation: `1 + 2 cosecx = -(sec^2 (x/2))/2`

Answer

To solve the equation \( 1 + 2 \csc x = -\frac{\sec^2 \left( \frac{x}{2} \right)}{2} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 1 + 2 \csc x = -\frac{\sec^2 \left( \frac{x}{2} \right)}{2} \]
We can rewrite \(\csc x\) and \(\sec^2 \left( \frac{x}{2} \right)\) in terms of sine and cosine:
\[ \csc x = \frac{1}{\sin x} \quad \text{and} \quad \sec^2 \left( \frac{x}{2} \right) = \frac{1}{\cos^2 \left( \frac{x}{2} \right)} \]
Thus, the equation becomes:
\[ 1 + \frac{2}{\sin x} = -\frac{1}{2 \cos^2 \left( \frac{x}{2} \right)} \]

### Step 2: Clear the fractions
To eliminate the fractions, we can multiply through by \(2 \sin x \cos^2 \left( \frac{x}{2} \right)\):
\[ 2 \sin x \cos^2 \left( \frac{x}{2} \right) + 4 \cos^2 \left( \frac{x}{2} \right) = -\sin x \]

### Step 3: Rearrange the equation
Rearranging gives:
\[ 2 \sin x \cos^2 \left( \frac{x}{2} \right) + \sin x + 4 \cos^2 \left( \frac{x}{2} \right) = 0 \]
Factoring out \(\sin x\):
\[ \sin x (2 \cos^2 \left( \frac{x}{2} \right) + 1) + 4 \cos^2 \left( \frac{x}{2} \right) = 0 \]

### Step 4: Set factors to zero
This gives us two cases to consider:
1. \(\sin x = 0\)
2. \(2 \cos^2 \left( \frac{x}{2} \right) + 1 + 4 \cos^2 \left( \frac{x}{2} \right) = 0\)

### Step 5: Solve \(\sin x = 0\)
The solutions for \(\sin x = 0\) are:
\[ x = n\pi \quad (n \in \mathbb{Z}) \]

### Step 6: Solve the second equation
For the second case:
\[ 6 \cos^2 \left( \frac{x}{2} \right) + 1 = 0 \]
This implies:
\[ \cos^2 \left( \frac{x}{2} \right) = -\frac{1}{6} \]
Since \(\cos^2\) cannot be negative, there are no solutions from this case.

### Final Solution
Thus, the only solutions to the original equation are:
\[ x = n\pi \quad (n \in \mathbb{Z}) \]

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