`sin(2"sin"^(-1)sqrt((63)/(65)))` is equal to

Options

A. `(2sqrt(126))/(65)`
B. `(4sqrt(65))/(65)`
C. `(8sqrt(63))/(65)`
D. `(sqrt(63))/(65)`

Explanation

To solve the expression \( \sin(2 \sin^{-1}(\sqrt{\frac{63}{65}})) \), we can follow these steps:

### Step 1: Let \( x = \sin^{-1}(\sqrt{\frac{63}{65}}) \)
This implies that \( \sin(x) = \sqrt{\frac{63}{65}} \).

### Step 2: Use the double angle formula for sine
The double angle formula for sine states that:
\[ \sin(2x) = 2 \sin(x) \cos(x) \]
So, we can write:
\[ \sin(2 \sin^{-1}(\sqrt{\frac{63}{65}})) = 2 \sin(x) \cos(x) \]

### Step 3: Calculate \( \cos(x) \)
Using the Pythagorean identity, we know that:
\[ \cos^2(x) + \sin^2(x) = 1 \]
Substituting \( \sin(x) = \sqrt{\frac{63}{65}} \):
\[ \cos^2(x) + \left(\sqrt{\frac{63}{65}}\right)^2 = 1 \]
\[ \cos^2(x) + \frac{63}{65} = 1 \]
\[ \cos^2(x) = 1 - \frac{63}{65} = \frac{2}{65} \]
Thus,
\[ \cos(x) = \sqrt{\frac{2}{65}} = \frac{\sqrt{2}}{\sqrt{65}} \]

### Step 4: Substitute \( \sin(x) \) and \( \cos(x) \) into the double angle formula
Now we can substitute back into the double angle formula:
\[ \sin(2x) = 2 \sin(x) \cos(x) = 2 \left(\sqrt{\frac{63}{65}}\right) \left(\frac{\sqrt{2}}{\sqrt{65}}\right) \]
\[ = 2 \cdot \sqrt{\frac{63 \cdot 2}{65 \cdot 65}} = 2 \cdot \sqrt{\frac{126}{4225}} \]

### Step 5: Simplify the expression
Now we simplify:
\[ = \frac{2\sqrt{126}}{65} \]

### Final Answer
Thus, the value of \( \sin(2 \sin^{-1}(\sqrt{\frac{63}{65}})) \) is:
\[ \frac{2\sqrt{126}}{65} \]