Can there exist triangles ABC satisfying the following relation? Write yes or no giving reasons:
`sinA+sin B=(sqrt3+1)/2, cos A cos B=(sqrt3)/4 = sin A sin B`.

Answer

To determine if there exist triangles ABC satisfying the given relations, we will analyze the equations step by step.

### Step 1: Analyze the first equation
We have the equation:
\[ \sin A + \sin B = \frac{\sqrt{3} + 1}{2} \]
To check if this can hold for angles A and B, we will find the maximum value of \(\sin A + \sin B\).

**Hint:** The maximum value of \(\sin A + \sin B\) occurs when both angles A and B are \(90^\circ\).

### Step 2: Calculate the maximum value of \(\sin A + \sin B\)
The maximum value of \(\sin A + \sin B\) is:
\[ \sin A + \sin B \leq \sin 90^\circ + \sin 90^\circ = 1 + 1 = 2 \]
Now we need to check if \(\frac{\sqrt{3} + 1}{2} \leq 2\).

**Hint:** Simplify \(\frac{\sqrt{3} + 1}{2}\) to see if it is less than or equal to 2.

### Step 3: Simplify \(\frac{\sqrt{3} + 1}{2}\)
Calculating:
\[ \sqrt{3} \approx 1.732 \implies \frac{\sqrt{3} + 1}{2} \approx \frac{1.732 + 1}{2} = \frac{2.732}{2} \approx 1.366 \]
Since \(1.366 < 2\), the first equation can hold.

**Hint:** Check if there are specific angles A and B that satisfy this equation.

### Step 4: Check specific angles for the first equation
Let’s try \(A = 60^\circ\) and \(B = 30^\circ\):
\[ \sin 60^\circ + \sin 30^\circ = \frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{\sqrt{3} + 1}{2} \]
This satisfies the first equation.

**Hint:** Now, we need to check the second equation.

### Step 5: Analyze the second equation
The second equation is:
\[ \cos A \cos B = \frac{\sqrt{3}}{4} = \sin A \sin B \]
We will first check \(\cos A \cos B\) for \(A = 60^\circ\) and \(B = 30^\circ\).

**Hint:** Calculate \(\cos 60^\circ\) and \(\cos 30^\circ\).

### Step 6: Calculate \(\cos A \cos B\)
Calculating:
\[ \cos 60^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Thus,
\[ \cos A \cos B = \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \]
This satisfies the first part of the second equation.

**Hint:** Now, we need to check \(\sin A \sin B\).

### Step 7: Calculate \(\sin A \sin B\)
Calculating:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2} \]
Thus,
\[ \sin A \sin B = \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{4} \]
This satisfies the second part of the second equation.

### Conclusion
Since both equations are satisfied with angles \(A = 60^\circ\), \(B = 30^\circ\), and \(C = 90^\circ\), we conclude that:

**Answer:** Yes, there exists a triangle ABC satisfying the given relations.