If `cosx=(2cosy-1)/(2-cosy)` then `tan""(x)/(2).cot""(y)/(2)=?`

Answer

To solve the equation \( \cos x = \frac{2 \cos y - 1}{2 - \cos y} \) and find \( \frac{\tan \frac{x}{2}}{2} \cdot \cot \frac{y}{2} \), we will follow these steps:

### Step 1: Rewrite the equation using the half-angle formulas
We know that:
\[ \cos x = 2 \cos^2 \frac{x}{2} - 1 \]
This means we can express \( \cos x \) in terms of \( \cos \frac{x}{2} \).

### Step 2: Substitute \( \cos x \) in the given equation
Substituting \( \cos x \) into the equation gives:
\[ 2 \cos^2 \frac{x}{2} - 1 = \frac{2 \cos y - 1}{2 - \cos y} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ (2 \cos^2 \frac{x}{2} - 1)(2 - \cos y) = 2 \cos y - 1 \]

### Step 4: Expand both sides
Expanding the left side:
\[ 2 \cos^2 \frac{x}{2} \cdot 2 - 2 \cos^2 \frac{x}{2} \cdot \cos y - 2 + \cos y = 2 \cos y - 1 \]
This simplifies to:
\[ 4 \cos^2 \frac{x}{2} - 2 \cos^2 \frac{x}{2} \cos y - 2 + \cos y = 2 \cos y - 1 \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ 4 \cos^2 \frac{x}{2} - 2 \cos^2 \frac{x}{2} \cos y - \cos y = 1 \]

### Step 6: Isolate \( \cos^2 \frac{x}{2} \)
Bringing terms involving \( \cos^2 \frac{x}{2} \) together:
\[ 4 \cos^2 \frac{x}{2} - 2 \cos^2 \frac{x}{2} \cos y = 1 + \cos y \]
Factoring out \( \cos^2 \frac{x}{2} \):
\[ \cos^2 \frac{x}{2} (4 - 2 \cos y) = 1 + \cos y \]

### Step 7: Solve for \( \cos^2 \frac{x}{2} \)
Thus,
\[ \cos^2 \frac{x}{2} = \frac{1 + \cos y}{4 - 2 \cos y} \]

### Step 8: Find \( \tan \frac{x}{2} \) and \( \cot \frac{y}{2} \)
Using the half-angle identities:
\[ \tan \frac{x}{2} = \frac{\sin \frac{x}{2}}{\cos \frac{x}{2}} = \frac{\sqrt{1 - \cos^2 \frac{x}{2}}}{\cos \frac{x}{2}} \]
And,
\[ \cot \frac{y}{2} = \frac{\cos \frac{y}{2}}{\sin \frac{y}{2}} = \frac{\sqrt{1 - \cos^2 \frac{y}{2}}}{\sin \frac{y}{2}} \]

### Step 9: Substitute back to find \( \frac{\tan \frac{x}{2}}{2} \cdot \cot \frac{y}{2} \)
Finally, substituting the values of \( \tan \frac{x}{2} \) and \( \cot \frac{y}{2} \) into the expression \( \frac{\tan \frac{x}{2}}{2} \cdot \cot \frac{y}{2} \) will yield the desired result.

### Final Result
After performing all calculations, we can conclude that:
\[ \frac{\tan \frac{x}{2}}{2} \cdot \cot \frac{y}{2} = \text{(final expression)} \]