If `sinbeta` is the harmonic mean of `sinalpha` and `cosalpha`, and `sintheta ` is the arithmetic mean of `sinalpha" and "cosalpha`, then which of the following is/are correct?
1. `sqrt(2)sin(alpha+(pi)/(4))sinbeta=sin2alpha`
2.`sqrt(2)sintheta=cos(alpha-(pi)/(4))`
Select the correct answer using the code given below:

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To solve the problem, we need to analyze the given conditions and verify both statements step by step.

### Given:
1. \( \sin \beta \) is the harmonic mean of \( \sin \alpha \) and \( \cos \alpha \).
2. \( \sin \theta \) is the arithmetic mean of \( \sin \alpha \) and \( \cos \alpha \).

### Step 1: Find \( \sin \beta \)
The harmonic mean \( H \) of two numbers \( a \) and \( b \) is given by:
\[ H = \frac{2ab}{a + b} \]
Here, \( a = \sin \alpha \) and \( b = \cos \alpha \). Therefore, we have:
\[ \sin \beta = \frac{2 \sin \alpha \cos \alpha}{\sin \alpha + \cos \alpha} \]

### Step 2: Find \( \sin \theta \)
The arithmetic mean \( A \) of two numbers \( a \) and \( b \) is given by:
\[ A = \frac{a + b}{2} \]
Thus, we have:
\[ \sin \theta = \frac{\sin \alpha + \cos \alpha}{2} \]

### Step 3: Analyze the first statement
The first statement is:
\[ \sqrt{2} \sin\left(\alpha + \frac{\pi}{4}\right) \sin \beta = \sin 2\alpha \]
Using the sine addition formula:
\[ \sin\left(\alpha + \frac{\pi}{4}\right) = \sin \alpha \cos\left(\frac{\pi}{4}\right) + \cos \alpha \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}(\sin \alpha + \cos \alpha) \]
Thus, we can rewrite the left side:
\[ \sqrt{2} \cdot \frac{1}{\sqrt{2}}(\sin \alpha + \cos \alpha) \sin \beta = (\sin \alpha + \cos \alpha) \sin \beta \]
Substituting \( \sin \beta \):
\[ (\sin \alpha + \cos \alpha) \cdot \frac{2 \sin \alpha \cos \alpha}{\sin \alpha + \cos \alpha} = 2 \sin \alpha \cos \alpha = \sin 2\alpha \]
Thus, the first statement is **true**.

### Step 4: Analyze the second statement
The second statement is:
\[ \sqrt{2} \sin \theta = \cos\left(\alpha - \frac{\pi}{4}\right) \]
Using the cosine subtraction formula:
\[ \cos\left(\alpha - \frac{\pi}{4}\right) = \cos \alpha \cos\left(\frac{\pi}{4}\right) + \sin \alpha \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}(\cos \alpha + \sin \alpha) \]
Thus, we can rewrite the right side:
\[ \sqrt{2} \sin \theta = \sqrt{2} \cdot \frac{\sin \alpha + \cos \alpha}{2} = \frac{\sqrt{2}}{2}(\sin \alpha + \cos \alpha) \]
Now substituting:
\[ \cos\left(\alpha - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}(\cos \alpha + \sin \alpha) \]
Thus,
\[ \sqrt{2} \sin \theta = \cos\left(\alpha - \frac{\pi}{4}\right) \]
This statement is also **true**.

### Conclusion
Both statements are correct.

### Final Answer
Both statements are true. The correct answer is option C: Both 1 and 2.

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