What is `sin^(-1)""(4)/(5)+sin^(-1)""(3)/(3)` equal to ?

Options

A. `(pi)/(2)`
B. `(pi)/(3)`
C. `(pi)/(4)`
D. `(pi)/(6)`

Answer

To solve the problem \( \sin^{-1}\left(\frac{4}{5}\right) + \sin^{-1}\left(\frac{3}{5}\right) \), we can follow these steps:

### Step 1: Define the angles
Let \( \theta = \sin^{-1}\left(\frac{3}{5}\right) \). This means that:
\[ \sin \theta = \frac{3}{5} \]

### Step 2: Find \( \cos \theta \)
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can find \( \cos \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]
Thus,
\[ \cos \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

### Step 3: Rewrite the expression
Now we can rewrite the original expression:
\[ \sin^{-1}\left(\frac{4}{5}\right) + \sin^{-1}\left(\frac{3}{5}\right) = \sin^{-1}\left(\frac{4}{5}\right) + \theta \]

### Step 4: Use the identity
We know that:
\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]
Thus, since \( \sin^{-1}\left(\frac{4}{5}\right) = \cos^{-1}\left(\frac{3}{5}\right) \) (because \( \cos \theta = \frac{4}{5} \)), we can write:
\[ \sin^{-1}\left(\frac{4}{5}\right) + \sin^{-1}\left(\frac{3}{5}\right) = \frac{\pi}{2} \]

### Conclusion
Therefore, the value of \( \sin^{-1}\left(\frac{4}{5}\right) + \sin^{-1}\left(\frac{3}{5}\right) \) is:
\[ \frac{\pi}{2} \]

### Final Answer
The answer is \( \frac{\pi}{2} \).

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