In `Delta ABC,if cos A+sin A-2/(cosB+sin B)=0,` then the value of `((a+b)/c)^4` is

Explanation

To solve the problem, we need to analyze the given equation and find the value of \(\left(\frac{a+b}{c}\right)^4\).

### Step-by-Step Solution:

1. **Given Equation:**
\[ \cos A + \sin A - \frac{2}{\cos B + \sin B} = 0 \]
Rearranging gives:
\[ \cos A + \sin A = \frac{2}{\cos B + \sin B} \]

2. **Cross-Multiplying:**
Multiply both sides by \((\cos B + \sin B)\):
\[ (\cos A + \sin A)(\cos B + \sin B) = 2 \]

3. **Expanding the Left Side:**
Using the distributive property:
\[ \cos A \cos B + \cos A \sin B + \sin A \cos B + \sin A \sin B = 2 \]

4. **Using Trigonometric Identities:**
Recognize that:
\[ \cos A \cos B + \sin A \sin B = \cos(A - B) \]
and
\[ \cos A \sin B + \sin A \cos B = \sin(A + B) \]
Therefore, we can rewrite the equation as:
\[ \cos(A - B) + \sin(A + B) = 2 \]

5. **Analyzing the Equation:**
The maximum value of \(\cos\) and \(\sin\) functions is 1. Thus, for the equation to hold:
\[ \cos(A - B) = 1 \quad \text{and} \quad \sin(A + B) = 1 \]

6. **Finding Angles:**
From \(\cos(A - B) = 1\), we have:
\[ A - B = 0 \implies A = B \]
From \(\sin(A + B) = 1\), we have:
\[ A + B = 90^\circ \]

7. **Finding Values of Angles:**
Since \(A = B\), we can substitute:
\[ 2A = 90^\circ \implies A = 45^\circ \quad \text{and} \quad B = 45^\circ \]

8. **Using the Sine Rule:**
In triangle \(ABC\), we know:
\[ C = 180^\circ - (A + B) = 180^\circ - 90^\circ = 90^\circ \]
Thus, triangle \(ABC\) is a right triangle with \(A = B = 45^\circ\).

9. **Finding Side Lengths:**
Let the lengths of sides opposite to angles \(A\) and \(B\) be \(a\) and \(b\) respectively. Since \(A = B\), we have \(a = b\). The hypotenuse \(c\) can be calculated using the Pythagorean theorem:
\[ c = \sqrt{a^2 + b^2} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

10. **Finding \(\frac{a+b}{c}\):**
Since \(a = b\):
\[ \frac{a+b}{c} = \frac{a + a}{a\sqrt{2}} = \frac{2a}{a\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

11. **Calculating \(\left(\frac{a+b}{c}\right)^4\):**
\[ \left(\frac{a+b}{c}\right)^4 = (\sqrt{2})^4 = 2^2 = 4 \]

### Final Answer:
\[ \left(\frac{a+b}{c}\right)^4 = 4 \]