Maximum value of `cosx (sinx +cos x)` is equal to :

Options

A. `sqrt(2)`
B. 2
C. `(sqrt(2)+1)/(2)`
D. `sqrt(2)+1`

Answer

To find the maximum value of the expression \( \cos x (\sin x + \cos x) \), we can follow these steps:

### Step 1: Rewrite the expression
Start with the expression:
\[ y = \cos x (\sin x + \cos x) \]
Distributing \( \cos x \) gives:
\[ y = \cos x \sin x + \cos^2 x \]

### Step 2: Use trigonometric identities
Recall that \( 2 \sin x \cos x = \sin 2x \) and \( \cos^2 x = \frac{1 + \cos 2x}{2} \). Thus, we can rewrite \( y \):
\[ y = \frac{1}{2} \sin 2x + \frac{1 + \cos 2x}{2} \]
This simplifies to:
\[ y = \frac{1}{2} (\sin 2x + 1 + \cos 2x) \]

### Step 3: Combine terms
Now, we can express the combined terms:
\[ y = \frac{1}{2} (\sin 2x + \cos 2x + 1) \]

### Step 4: Find the maximum value of \( \sin 2x + \cos 2x \)
The maximum value of \( \sin 2x + \cos 2x \) can be found using the formula for the maximum value of \( a \sin \theta + b \cos \theta \), which is \( \sqrt{a^2 + b^2} \). Here, \( a = 1 \) and \( b = 1 \):
\[ \text{Maximum value of } \sin 2x + \cos 2x = \sqrt{1^2 + 1^2} = \sqrt{2} \]

### Step 5: Substitute back to find the maximum value of \( y \)
Now substitute this maximum value back into the equation for \( y \):
\[ y_{\text{max}} = \frac{1}{2} (\sqrt{2} + 1) \]

### Final Answer
Thus, the maximum value of \( \cos x (\sin x + \cos x) \) is:
\[ \frac{\sqrt{2} + 1}{2} \]