If x and y are the real numbers satisfying the equation `12sinx + 5cos x = 2y^2 - 8y +21`, then the value of `12cot((xy)/2)` is:

Answer

To solve the equation \( 12 \sin x + 5 \cos x = 2y^2 - 8y + 21 \) and find the value of \( 12 \cot\left(\frac{xy}{2}\right) \), we can follow these steps:

### Step 1: Analyze the Left-Hand Side (LHS)
The LHS is \( 12 \sin x + 5 \cos x \). We can express this in the form \( R \sin(x + \alpha) \).

### Step 2: Find \( R \) and \( \alpha \)
To find \( R \) and \( \alpha \), we use the identities:
- \( R = \sqrt{a^2 + b^2} \) where \( a = 12 \) and \( b = 5 \).
- \( \tan \alpha = \frac{b}{a} = \frac{5}{12} \).

Calculating \( R \):
\[ R = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13. \]

### Step 3: Rewrite LHS
Now we can rewrite the LHS:
\[ 12 \sin x + 5 \cos x = 13 \sin(x + \alpha). \]
Since \( \sin(x + \alpha) \) has a maximum value of 1, we have:
\[ 12 \sin x + 5 \cos x \leq 13. \]

### Step 4: Analyze the Right-Hand Side (RHS)
The RHS is \( 2y^2 - 8y + 21 \). We can complete the square:
\[ 2y^2 - 8y + 21 = 2(y^2 - 4y) + 21 = 2((y - 2)^2 - 4) + 21 = 2(y - 2)^2 + 13. \]
Since \( (y - 2)^2 \geq 0 \), we have:
\[ 2(y - 2)^2 + 13 \geq 13. \]

### Step 5: Set LHS Equal to RHS
For the equation to hold, we need:
\[ 12 \sin x + 5 \cos x = 2y^2 - 8y + 21 = 13. \]
This gives us two conditions:
1. \( 12 \sin x + 5 \cos x = 13 \)
2. \( 2(y - 2)^2 = 0 \) which implies \( y = 2 \).

### Step 6: Solve for \( x \)
From \( 12 \sin x + 5 \cos x = 13 \), we can deduce:
\[ \sin(x + \alpha) = 1 \implies x + \alpha = \frac{\pi}{2} \implies x = \frac{\pi}{2} - \alpha. \]
Where \( \alpha = \tan^{-1}\left(\frac{5}{12}\right) \).

### Step 7: Find \( \cot\left(\frac{xy}{2}\right) \)
Now, substituting \( y = 2 \):
\[ xy = x \cdot 2 = 2\left(\frac{\pi}{2} - \tan^{-1}\left(\frac{5}{12}\right)\right) = \pi - 2\tan^{-1}\left(\frac{5}{12}\right). \]
Thus,
\[ \frac{xy}{2} = \frac{\pi}{2} - \tan^{-1}\left(\frac{5}{12}\right). \]
Using the cotangent identity:
\[ \cot\left(\frac{\pi}{2} - \theta\right) = \tan(\theta), \]
we have:
\[ \cot\left(\frac{xy}{2}\right) = \tan\left(\tan^{-1}\left(\frac{5}{12}\right)\right) = \frac{5}{12}. \]

### Step 8: Calculate \( 12 \cot\left(\frac{xy}{2}\right) \)
Finally, we compute:
\[ 12 \cot\left(\frac{xy}{2}\right) = 12 \cdot \frac{5}{12} = 5. \]

### Final Answer
The value of \( 12 \cot\left(\frac{xy}{2}\right) \) is \( \boxed{5} \).