Simplify: cos^-1((sinx+cosx)/(sqrt(2)))

Explanation

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

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

### Step 2: Recognize the values of sine and cosine
Recall that:
\[ \frac{1}{\sqrt{2}} = \sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) \]
Thus, we can express \( \frac{1}{\sqrt{2}} \) as:
\[ \frac{1}{\sqrt{2}} = \cos\left(\frac{\pi}{4}\right) \text{ and } \sin\left(\frac{\pi}{4}\right) \]

### Step 3: Use angle addition formula
Using the angle addition formula for sine:
\[ \sin A + \cos A = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]
So we can write:
\[ \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]
Substituting this back into our expression gives:
\[ y = \cos^{-1}\left(\sin\left(x + \frac{\pi}{4}\right)\right) \]

### Step 4: Use the identity for cosine
Now, we know that:
\[ \cos^{-1}(\sin \theta) = \frac{\pi}{2} - \theta \quad \text{(for } 0 \leq \theta \leq \frac{\pi}{2}\text{)} \]
Thus, we can write:
\[ y = \frac{\pi}{2} - \left(x + \frac{\pi}{4}\right) \]

### Step 5: Simplify the final expression
Now, simplifying this gives:
\[ y = \frac{\pi}{2} - x - \frac{\pi}{4} = \frac{\pi}{4} - x \]

### Final Answer
Thus, the simplified expression is:
\[ \cos^{-1}\left(\frac{\sin x + \cos x}{\sqrt{2}}\right) = \frac{\pi}{4} - x \]