If A,B,C be all acute angles and `sin ( B+C-A) = cos ( C+A-B) = tan ( A+B-C ) =1`, then `A="………..",B="……….",C="…………….."`

Answer

To solve the problem, we need to find the values of acute angles A, B, and C given the equations:

1. \( \sin(B + C - A) = 1 \)
2. \( \cos(C + A - B) = 1 \)
3. \( \tan(A + B - C) = 1 \)

### Step-by-Step Solution:

**Step 1: Analyze the first equation**
From the equation \( \sin(B + C - A) = 1 \), we know that the sine function equals 1 at \( \frac{\pi}{2} \). Therefore, we can write:
\[ B + C - A = \frac{\pi}{2} \quad \text{(Equation 1)} \]

**Step 2: Analyze the second equation**
From the equation \( \cos(C + A - B) = 1 \), we know that the cosine function equals 1 at \( 0 \). Therefore, we can write:
\[ C + A - B = 0 \quad \text{(Equation 2)} \]

**Step 3: Analyze the third equation**
From the equation \( \tan(A + B - C) = 1 \), we know that the tangent function equals 1 at \( \frac{\pi}{4} \). Therefore, we can write:
\[ A + B - C = \frac{\pi}{4} \quad \text{(Equation 3)} \]

**Step 4: Solve the equations**
Now we have a system of three equations:

1. \( B + C - A = \frac{\pi}{2} \) (Equation 1)
2. \( C + A - B = 0 \) (Equation 2)
3. \( A + B - C = \frac{\pi}{4} \) (Equation 3)

**Step 5: Add Equation 1 and Equation 2**
Adding Equation 1 and Equation 2:
\[ (B + C - A) + (C + A - B) = \frac{\pi}{2} + 0 \]
This simplifies to:
\[ 2C = \frac{\pi}{2} \]
Thus, we find:
\[ C = \frac{\pi}{4} \]

**Step 6: Substitute C into Equation 2**
Now substitute \( C = \frac{\pi}{4} \) into Equation 2:
\[ \frac{\pi}{4} + A - B = 0 \]
This gives us:
\[ A - B = -\frac{\pi}{4} \quad \text{(Equation 4)} \]

**Step 7: Substitute C into Equation 3**
Now substitute \( C = \frac{\pi}{4} \) into Equation 3:
\[ A + B - \frac{\pi}{4} = \frac{\pi}{4} \]
This simplifies to:
\[ A + B = \frac{\pi}{2} \quad \text{(Equation 5)} \]

**Step 8: Solve Equations 4 and 5**
Now we have two equations:
1. \( A - B = -\frac{\pi}{4} \) (Equation 4)
2. \( A + B = \frac{\pi}{2} \) (Equation 5)

Adding these two equations:
\[ (A - B) + (A + B) = -\frac{\pi}{4} + \frac{\pi}{2} \]
This simplifies to:
\[ 2A = \frac{\pi}{4} \]
Thus, we find:
\[ A = \frac{\pi}{8} \]

**Step 9: Substitute A into Equation 5**
Now substitute \( A = \frac{\pi}{8} \) into Equation 5:
\[ \frac{\pi}{8} + B = \frac{\pi}{2} \]
This gives us:
\[ B = \frac{\pi}{2} - \frac{\pi}{8} = \frac{4\pi}{8} - \frac{\pi}{8} = \frac{3\pi}{8} \]

### Final Values:
- \( A = \frac{\pi}{8} \)
- \( B = \frac{3\pi}{8} \)
- \( C = \frac{\pi}{4} \)

### Conclusion:
Thus, the values of A, B, and C in degrees are:
- \( A = 22.5^\circ \)
- \( B = 67.5^\circ \)
- \( C = 45^\circ \)