`Sin^(-1)((12)/(13))-sin^(-1)((3)/(5))` is equal to

Options

A. `(pi)/(2)-cos^(-1)((9)/(65))`
B. `pi-cos^(-1)((33)/(65))`
C. `(pi)/(2)-sin^(-1)((56)/(65))`
D. `pi-sin^(-1)((63)/(65))`

Explanation

To solve the expression \( \sin^{-1}\left(\frac{12}{13}\right) - \sin^{-1}\left(\frac{3}{5}\right) \), 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 \( y \)
Let:
- \( x = \frac{12}{13} \)
- \( y = \frac{3}{5} \)

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

### Step 3: Calculate \( \sqrt{1 - x^2} \)
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 4: Substitute into the formula
Now, we substitute \( x \), \( y \), \( \sqrt{1 - y^2} \), and \( \sqrt{1 - x^2} \) into the formula:
\[ \sin^{-1}\left(\frac{12}{13} \cdot \frac{4}{5} - \frac{3}{5} \cdot \frac{5}{13}\right) \]

Calculating the terms:
\[ \frac{12}{13} \cdot \frac{4}{5} = \frac{48}{65} \]
\[ \frac{3}{5} \cdot \frac{5}{13} = \frac{15}{65} \]

Now, subtract these two results:
\[ \frac{48}{65} - \frac{15}{65} = \frac{33}{65} \]

### Step 5: Final result
Thus, we have:
\[ \sin^{-1}\left(\frac{33}{65}\right) \]

### Conclusion
The expression \( \sin^{-1}\left(\frac{12}{13}\right) - \sin^{-1}\left(\frac{3}{5}\right) \) simplifies to:
\[ \sin^{-1}\left(\frac{33}{65}\right) \]