The maximum value of `1+sin(pi/4+theta)+2cos(pi/4-theta)` for real values of `theta` is

Explanation

To find the maximum value of the expression \(1 + \sin\left(\frac{\pi}{4} + \theta\right) + 2\cos\left(\frac{\pi}{4} - \theta\right)\), we can follow these steps:

### Step 1: Rewrite the trigonometric functions
We can use the angle addition and subtraction formulas for sine and cosine:
- \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)
- \(\cos(a - b) = \cos a \cos b + \sin a \sin b\)

Applying these to our expression:
\[ \sin\left(\frac{\pi}{4} + \theta\right) = \sin\frac{\pi}{4} \cos\theta + \cos\frac{\pi}{4} \sin\theta \]
\[ \cos\left(\frac{\pi}{4} - \theta\right) = \cos\frac{\pi}{4} \cos\theta + \sin\frac{\pi}{4} \sin\theta \]

### Step 2: Substitute the values of \(\sin\frac{\pi}{4}\) and \(\cos\frac{\pi}{4}\)
Since \(\sin\frac{\pi}{4} = \cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}\), we can substitute these values:
\[ \sin\left(\frac{\pi}{4} + \theta\right) = \frac{1}{\sqrt{2}} \cos\theta + \frac{1}{\sqrt{2}} \sin\theta \]
\[ \cos\left(\frac{\pi}{4} - \theta\right) = \frac{1}{\sqrt{2}} \cos\theta + \frac{1}{\sqrt{2}} \sin\theta \]

### Step 3: Substitute back into the original expression
Now substituting these back into the original expression:
\[ 1 + \left(\frac{1}{\sqrt{2}} \cos\theta + \frac{1}{\sqrt{2}} \sin\theta\right) + 2\left(\frac{1}{\sqrt{2}} \cos\theta + \frac{1}{\sqrt{2}} \sin\theta\right) \]
This simplifies to:
\[ 1 + \frac{1}{\sqrt{2}} \cos\theta + \frac{1}{\sqrt{2}} \sin\theta + \frac{2}{\sqrt{2}} \cos\theta + \frac{2}{\sqrt{2}} \sin\theta \]
\[ = 1 + \left(\frac{1 + 2}{\sqrt{2}}\right) \cos\theta + \left(\frac{1 + 2}{\sqrt{2}}\right) \sin\theta \]
\[ = 1 + \frac{3}{\sqrt{2}} \cos\theta + \frac{3}{\sqrt{2}} \sin\theta \]

### Step 4: Factor out the common term
We can factor out \(\frac{3}{\sqrt{2}}\):
\[ = 1 + \frac{3}{\sqrt{2}} \left(\cos\theta + \sin\theta\right) \]

### Step 5: Use the identity for maximum value
The expression \(\cos\theta + \sin\theta\) can be rewritten using the identity:
\[ \cos\theta + \sin\theta = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \]
The maximum value of \(\sin\) function is 1, hence:
\[ \cos\theta + \sin\theta \leq \sqrt{2} \]

### Step 6: Substitute the maximum value back
Thus, the maximum value of our expression becomes:
\[ 1 + \frac{3}{\sqrt{2}} \cdot \sqrt{2} = 1 + 3 = 4 \]

### Final Answer
The maximum value of \(1 + \sin\left(\frac{\pi}{4} + \theta\right) + 2\cos\left(\frac{\pi}{4} - \theta\right)\) is \(\boxed{4}\).

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