`cos^(-1)((12)/(13))+sin^(-1)((3)/(5))`=

Options

A. `sin^(-1)((56)/(65))`
B. `sin^(-1)((16)/(65))`
C. `sin^(-1)((20)/(65))`
D. `sin^(-1)((36)/(65))`

Answer

To solve the expression \( \cos^{-1}\left(\frac{12}{13}\right) + \sin^{-1}\left(\frac{3}{5}\right) \), we can follow these steps:

### Step 1: Let \( \alpha = \cos^{-1}\left(\frac{12}{13}\right) \)
This means that \( \cos \alpha = \frac{12}{13} \).

### Step 2: Find \( \sin \alpha \)
We can use the Pythagorean identity:
\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]
Substituting \( \cos \alpha \):
\[ \sin^2 \alpha + \left(\frac{12}{13}\right)^2 = 1 \]
Calculating \( \left(\frac{12}{13}\right)^2 \):
\[ \sin^2 \alpha + \frac{144}{169} = 1 \]
\[ \sin^2 \alpha = 1 - \frac{144}{169} = \frac{169 - 144}{169} = \frac{25}{169} \]
Taking the square root:
\[ \sin \alpha = \frac{5}{13} \]

### Step 3: Rewrite the expression
Now we can rewrite the original expression:
\[ \cos^{-1}\left(\frac{12}{13}\right) + \sin^{-1}\left(\frac{3}{5}\right) = \alpha + \sin^{-1}\left(\frac{3}{5}\right) \]

### Step 4: Let \( \beta = \sin^{-1}\left(\frac{3}{5}\right) \)
This means that \( \sin \beta = \frac{3}{5} \).

### Step 5: Find \( \cos \beta \)
Using the Pythagorean identity again:
\[ \sin^2 \beta + \cos^2 \beta = 1 \]
Substituting \( \sin \beta \):
\[ \left(\frac{3}{5}\right)^2 + \cos^2 \beta = 1 \]
Calculating \( \left(\frac{3}{5}\right)^2 \):
\[ \frac{9}{25} + \cos^2 \beta = 1 \]
\[ \cos^2 \beta = 1 - \frac{9}{25} = \frac{25 - 9}{25} = \frac{16}{25} \]
Taking the square root:
\[ \cos \beta = \frac{4}{5} \]

### Step 6: Use the sine addition formula
Now, we can use the sine addition formula:
\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]
Substituting the values:
\[ \sin(\alpha + \beta) = \left(\frac{5}{13}\right) \left(\frac{4}{5}\right) + \left(\frac{12}{13}\right) \left(\frac{3}{5}\right) \]
Calculating:
\[ = \frac{20}{65} + \frac{36}{65} = \frac{56}{65} \]

### Step 7: Final result
Thus, we have:
\[ \sin(\alpha + \beta) = \frac{56}{65} \]
So,
\[ \cos^{-1}\left(\frac{12}{13}\right) + \sin^{-1}\left(\frac{3}{5}\right) = \sin^{-1}\left(\frac{56}{65}\right) \]

### Conclusion
The final answer is:
\[ \cos^{-1}\left(\frac{12}{13}\right) + \sin^{-1}\left(\frac{3}{5}\right) = \sin^{-1}\left(\frac{56}{65}\right) \]