The maximum value of `cos^2(45^0+x)+(sinx-cosx)^2` is ______

Explanation

To find the maximum value of the expression \( \cos^2(45^\circ + x) + (\sin x - \cos x)^2 \), we can break it down step by step.

### Step 1: Simplify \( \cos^2(45^\circ + x) \)
Using the cosine addition formula, we have:
\[ \cos(45^\circ + x) = \cos 45^\circ \cos x - \sin 45^\circ \sin x \]
Since \( \cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}} \), we can substitute:
\[ \cos(45^\circ + x) = \frac{1}{\sqrt{2}} \cos x - \frac{1}{\sqrt{2}} \sin x = \frac{1}{\sqrt{2}} (\cos x - \sin x) \]
Now, squaring this gives:
\[ \cos^2(45^\circ + x) = \left(\frac{1}{\sqrt{2}} (\cos x - \sin x)\right)^2 = \frac{1}{2} (\cos x - \sin x)^2 \]

### Step 2: Expand \( (\sin x - \cos x)^2 \)
Next, we expand \( (\sin x - \cos x)^2 \):
\[ (\sin x - \cos x)^2 = \sin^2 x - 2 \sin x \cos x + \cos^2 x \]
Using the identity \( \sin^2 x + \cos^2 x = 1 \):
\[ (\sin x - \cos x)^2 = 1 - 2 \sin x \cos x \]

### Step 3: Combine the two parts
Now we combine both parts:
\[ \cos^2(45^\circ + x) + (\sin x - \cos x)^2 = \frac{1}{2} (\cos x - \sin x)^2 + (1 - 2 \sin x \cos x) \]
Substituting \( (\cos x - \sin x)^2 = \cos^2 x - 2 \sin x \cos x + \sin^2 x = 1 - 2 \sin x \cos x \):
\[ = \frac{1}{2} (1 - 2 \sin x \cos x) + (1 - 2 \sin x \cos x) \]
\[ = \frac{1}{2} + 1 - 2 \sin x \cos x + 1 - 2 \sin x \cos x \]
\[ = \frac{1}{2} + 2 - 4 \sin x \cos x \]
\[ = \frac{5}{2} - 4 \sin x \cos x \]

### Step 4: Maximize the expression
We know that \( \sin x \cos x = \frac{1}{2} \sin(2x) \). The maximum value of \( \sin(2x) \) is 1, thus:
\[ \sin x \cos x \text{ has a maximum value of } \frac{1}{2} \]
Substituting this back into our expression:
\[ \frac{5}{2} - 4 \cdot \frac{1}{2} = \frac{5}{2} - 2 = \frac{1}{2} \]

### Step 5: Conclusion
Thus, the maximum value of the expression \( \cos^2(45^\circ + x) + (\sin x - \cos x)^2 \) is:
\[ \boxed{3} \]