The value of `sin^(-1)((12)/(13)) - sin ^(-1)((3)/(5))` is equal to

Options

A. `pi-sin ^(-1) ((63)/(65))`
B. `(pi)/(2) - sin ^(-1)((56)/(65))`
C. `(pi)/(2) - cos ^(-1)((9)/(65)) `
D. `pi - cos ^(-1)((3)/(65))`

Explanation

To solve the problem \( \sin^{-1}\left(\frac{12}{13}\right) - \sin^{-1}\left(\frac{3}{5}\right) \), we can use the formula for the difference of two inverse sine functions:

\[ \sin^{-1}(x) - \sin^{-1}(y) = \sin^{-1}\left(x\sqrt{1-y^2} - y\sqrt{1-x^2}\right) \]

### Step 1: Identify \( x \) and \( y \)
Here, we have:
- \( x = \frac{12}{13} \)
- \( y = \frac{3}{5} \)

### Step 2: Calculate \( \sqrt{1 - y^2} \) and \( \sqrt{1 - x^2} \)
First, we calculate \( \sqrt{1 - y^2} \):
\[ y^2 = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]
\[ 1 - y^2 = 1 - \frac{9}{25} = \frac{25 - 9}{25} = \frac{16}{25} \]
\[ \sqrt{1 - y^2} = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

Next, we calculate \( \sqrt{1 - x^2} \):
\[ x^2 = \left(\frac{12}{13}\right)^2 = \frac{144}{169} \]
\[ 1 - x^2 = 1 - \frac{144}{169} = \frac{169 - 144}{169} = \frac{25}{169} \]
\[ \sqrt{1 - x^2} = \sqrt{\frac{25}{169}} = \frac{5}{13} \]

### Step 3: Substitute into the formula
Now we substitute into the formula:
\[ \sin^{-1}\left(\frac{12}{13}\right) - \sin^{-1}\left(\frac{3}{5}\right) = \sin^{-1}\left(\frac{12}{13} \cdot \frac{4}{5} - \frac{3}{5} \cdot \frac{5}{13}\right) \]

### Step 4: Calculate the expression inside the inverse sine
Calculating the terms:
\[ \frac{12}{13} \cdot \frac{4}{5} = \frac{48}{65} \]
\[ \frac{3}{5} \cdot \frac{5}{13} = \frac{15}{65} \]

Now, substituting these back:
\[ \sin^{-1}\left(\frac{48}{65} - \frac{15}{65}\right) = \sin^{-1}\left(\frac{33}{65}\right) \]

### Final Answer
Thus, the value of \( \sin^{-1}\left(\frac{12}{13}\right) - \sin^{-1}\left(\frac{3}{5}\right) \) is:
\[ \sin^{-1}\left(\frac{33}{65}\right) \]