simplify `cos^(-1)``((sinx+cosx)/sqrt2)```,`pi/4`< x <`(5pi )/4`

Explanation

To simplify the expression \( \cos^{-1} \left( \frac{\sin x + \cos x}{\sqrt{2}} \right) \) for \( \frac{\pi}{4} < x < \frac{5\pi}{4} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \cos^{-1} \left( \frac{\sin x + \cos x}{\sqrt{2}} \right) \]

### Step 2: Recognize the trigonometric identity
Notice that:
\[ \sin x + \cos x = \sqrt{2} \left( \sin x \cdot \frac{1}{\sqrt{2}} + \cos x \cdot \frac{1}{\sqrt{2}} \right) \]
This can be rewritten using the sine addition formula:
\[ \sin x + \cos x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \]

### Step 3: Substitute into the expression
Now, substitute this into the original expression:
\[ \frac{\sin x + \cos x}{\sqrt{2}} = \sin \left( x + \frac{\pi}{4} \right) \]
Thus, we have:
\[ \cos^{-1} \left( \sin \left( x + \frac{\pi}{4} \right) \right) \]

### Step 4: Use the identity for inverse cosine
Using the identity \( \cos^{-1}(\sin \theta) = \frac{\pi}{2} - \theta \) when \( \theta \) is in the appropriate range, we can write:
\[ \cos^{-1} \left( \sin \left( x + \frac{\pi}{4} \right) \right) = \frac{\pi}{2} - \left( x + \frac{\pi}{4} \right) \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ \frac{\pi}{2} - \left( x + \frac{\pi}{4} \right) = \frac{\pi}{2} - x - \frac{\pi}{4} = \frac{\pi}{4} - x \]

### Final Result
Thus, we have:
\[ \cos^{-1} \left( \frac{\sin x + \cos x}{\sqrt{2}} \right) = \frac{\pi}{4} - x \]