In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ?
1. **Identify the properties of the parallelogram and the given conditions:**
   - In parallelogram \(ABCD\), the diagonals \(AC\) and \(BD\) intersect at their midpoints.
   - Given \(AC = 10\) and \(BD = 28\).
   - Points \(K\) and \(L\) move such that \(AK = BD = 28\) and \(BL = AC = 10\).

2. **Determine the coordinates of the midpoints \(M\) and \(N\):**
   - Let \(O\) be the intersection of the diagonals \(AC\) and \(BD\). Since \(O\) is the midpoint of both diagonals, \(O\) divides \(AC\) and \(BD\) into equal segments.
   - Therefore, \(AO = OC = \frac{AC}{2} = 5\) and \(BO = OD = \frac{BD}{2} = 14\).

3. **Analyze the movement of points \(K\) and \(L\):**
   - Since \(AK = 28\), point \(K\) lies on a circle centered at \(A\) with radius 28.
   - Similarly, since \(BL = 10\), point \(L\) lies on a circle centered at \(B\) with radius 10.

4. **Find the midpoints \(M\) and \(N\):**
   - \(M\) is the midpoint of \(CK\), and \(N\) is the midpoint of \(DL\).

5. **Calculate the angles \(\angle BMD\) and \(\angle ANC\):**
   - Since \(M\) and \(N\) are midpoints, they lie on the lines connecting the midpoints of the sides of the parallelogram to the points \(K\) and \(L\).

6. **Use trigonometric identities to find the required value:**
   - We need to find the maximum value of \(\cot^2 \left(\frac{\angle BMD}{2}\right) + \tan^2 \left(\frac{\angle ANC}{2}\right)\).

7. **Simplify the trigonometric expressions:**
   - Using the identity \(\cot^2 x + \tan^2 x = \frac{1}{\sin^2 x \cos^2 x}\), we can simplify the expression.

8. **Determine the maximum value:**
   - Since \(\cot^2 x + \tan^2 x\) is always greater than or equal to 2, the maximum value is achieved when \(\sin^2 x \cos^2 x\) is minimized.

9. **Conclude the solution:**
   - The maximum value of \(\cot^2 \left(\frac{\angle BMD}{2}\right) + \tan^2 \left(\frac{\angle ANC}{2}\right)\) is 2.

The final answer is \(\boxed{2}\).