Find the value of `sin^(-1) ((3)/(5)) + tan^(-1) ((1)/(7))`

Explanation

To solve the problem \( \sin^{-1}\left(\frac{3}{5}\right) + \tan^{-1}\left(\frac{1}{7}\right) \), we can follow these steps:

### Step 1: Define the angle for \( \sin^{-1}\left(\frac{3}{5}\right) \)

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 \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]
Thus, \( \cos \theta = \frac{4}{5} \) (since \( \theta \) is in the first quadrant).

### Step 3: Find \( \tan \theta \)

Now, we can find \( \tan \theta \):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]

### Step 4: Rewrite the expression

Now we can rewrite the original expression:
\[ \sin^{-1}\left(\frac{3}{5}\right) + \tan^{-1}\left(\frac{1}{7}\right) = \theta + \tan^{-1}\left(\frac{1}{7}\right) \]

### Step 5: Use the formula for \( \tan^{-1}(x) + \tan^{-1}(y) \)

We can use the formula:
\[ \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \]
where \( x = \tan \theta = \frac{3}{4} \) and \( y = \frac{1}{7} \).

### Step 6: Calculate \( \tan^{-1}\left(\frac{3/4 + 1/7}{1 - (3/4)(1/7)}\right) \)

Calculating the numerator:
\[ \frac{3}{4} + \frac{1}{7} = \frac{21 + 4}{28} = \frac{25}{28} \]

Calculating the denominator:
\[ 1 - \left(\frac{3}{4} \cdot \frac{1}{7}\right) = 1 - \frac{3}{28} = \frac{28 - 3}{28} = \frac{25}{28} \]

### Step 7: Combine the results

Now we can substitute back into the formula:
\[ \tan^{-1}\left(\frac{\frac{25}{28}}{\frac{25}{28}}\right) = \tan^{-1}(1) \]

### Step 8: Find the final value

Since \( \tan^{-1}(1) = \frac{\pi}{4} \), we have:
\[ \sin^{-1}\left(\frac{3}{5}\right) + \tan^{-1}\left(\frac{1}{7}\right) = \frac{\pi}{4} \]

### Final Answer
\[ \sin^{-1}\left(\frac{3}{5}\right) + \tan^{-1}\left(\frac{1}{7}\right) = \frac{\pi}{4} \]