What is `sqrt(1+sin2theta)` equal to ?

Options

A. `costheta-sintheta`
B. `costheta+sintheta`
C. `2costheta+sintheta`
D. `costheta+2sintheta`

Explanation

To solve the expression \( \sqrt{1 + \sin 2\theta} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x = \sqrt{1 + \sin 2\theta} \]

### Step 2: Use the double angle identity
Recall that the double angle identity for sine states that:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Substituting this into our expression gives:
\[ x = \sqrt{1 + 2 \sin \theta \cos \theta} \]

### Step 3: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, we can rewrite \(1\) as \(\sin^2 \theta + \cos^2 \theta\):
\[ x = \sqrt{\sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta} \]

### Step 4: Recognize the perfect square
The expression inside the square root can be recognized as a perfect square:
\[ \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta = (\sin \theta + \cos \theta)^2 \]
So we can rewrite \(x\) as:
\[ x = \sqrt{(\sin \theta + \cos \theta)^2} \]

### Step 5: Simplify the square root
Taking the square root gives us:
\[ x = \sin \theta + \cos \theta \]
(Note: We assume \( \sin \theta + \cos \theta \geq 0 \) for the principal square root.)

### Final Answer
Thus, we conclude that:
\[ \sqrt{1 + \sin 2\theta} = \sin \theta + \cos \theta \]

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