If `sinalpha+cosalpha=p`, then what is `cos^(2)(2alpha)` equal to ?

Options

A. `p^(2)`
B. `p^(2)-1`
C. `p^(2)(2-p^(2))`
D. `p^(2)+1`

Explanation

To solve the problem where \( \sin \alpha + \cos \alpha = p \) and we need to find \( \cos^2(2\alpha) \), we can follow these steps:

### Step 1: Square the given equation
We start with the equation:
\[ \sin \alpha + \cos \alpha = p \]
Now, we square both sides:
\[ (\sin \alpha + \cos \alpha)^2 = p^2 \]

### Step 2: Expand the left side
Expanding the left side, we get:
\[ \sin^2 \alpha + 2\sin \alpha \cos \alpha + \cos^2 \alpha = p^2 \]
Using the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \), we can substitute:
\[ 1 + 2\sin \alpha \cos \alpha = p^2 \]

### Step 3: Express \( \sin 2\alpha \)
We know that \( 2\sin \alpha \cos \alpha = \sin 2\alpha \), so we can rewrite the equation as:
\[ 1 + \sin 2\alpha = p^2 \]
This gives us:
\[ \sin 2\alpha = p^2 - 1 \]

### Step 4: Find \( \cos^2(2\alpha) \)
We use the identity \( \cos^2(2\alpha) = 1 - \sin^2(2\alpha) \). First, we need to find \( \sin^2(2\alpha) \):
\[ \sin^2(2\alpha) = (p^2 - 1)^2 \]

### Step 5: Substitute into the cosine identity
Now substituting this into the cosine identity:
\[ \cos^2(2\alpha) = 1 - (p^2 - 1)^2 \]

### Step 6: Expand and simplify
Expanding \( (p^2 - 1)^2 \):
\[ (p^2 - 1)^2 = p^4 - 2p^2 + 1 \]
Thus, we have:
\[ \cos^2(2\alpha) = 1 - (p^4 - 2p^2 + 1) \]
This simplifies to:
\[ \cos^2(2\alpha) = 1 - p^4 + 2p^2 - 1 \]
\[ \cos^2(2\alpha) = 2p^2 - p^4 \]

### Step 7: Factor the expression
We can factor this expression:
\[ \cos^2(2\alpha) = p^2(2 - p^2) \]

### Final Answer
Thus, the final answer is:
\[ \cos^2(2\alpha) = p^2(2 - p^2) \]

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