If `A=2 tan^(-1) (2sqrt2-1) and B=3 sin^(-1)(1/3) + sin^(-1) (3/5)`, then

Options

A. `A gt B`
B. `A lt B`
C. `A=B`
D. none

Answer

To solve the problem, we need to evaluate the expressions for \( A \) and \( B \) given by:

\[ A = 2 \tan^{-1}(2\sqrt{2} - 1) \]
\[ B = 3 \sin^{-1}\left(\frac{1}{3}\right) + \sin^{-1}\left(\frac{3}{5}\right) \]

### Step 1: Evaluate \( A \)

1. **Calculate \( \tan^{-1}(2\sqrt{2} - 1) \)**:
- Let \( x = 2\sqrt{2} - 1 \).
- We need to find \( \tan^{-1}(x) \).

2. **Approximate \( x \)**:
- \( 2\sqrt{2} \approx 2 \times 1.414 = 2.828 \)
- So, \( x \approx 2.828 - 1 = 1.828 \).

3. **Compare \( x \) with \( \sqrt{3} \)**:
- \( \sqrt{3} \approx 1.732 \).
- Since \( 1.828 > 1.732 \), we have \( \tan^{-1}(1.828) > \tan^{-1}(\sqrt{3}) \).

4. **Find \( \tan^{-1}(\sqrt{3}) \)**:
- \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).

5. **Thus, \( \tan^{-1}(2\sqrt{2} - 1) > \frac{\pi}{3} \)**:
- Therefore, \( 2 \tan^{-1}(2\sqrt{2} - 1) > 2 \cdot \frac{\pi}{3} = \frac{2\pi}{3} \).

So, we conclude that:

\[ A > \frac{2\pi}{3} \]

### Step 2: Evaluate \( B \)

1. **Calculate \( 3 \sin^{-1}\left(\frac{1}{3}\right) \)**:
- Let \( y = \sin^{-1}\left(\frac{1}{3}\right) \).
- We know \( \sin(y) = \frac{1}{3} \).

2. **Use the triple angle formula for sine**:
- \( 3 \sin^{-1}(x) = \sin^{-1}(3x - 4x^3) \).
- Here, \( x = \frac{1}{3} \).
- Calculate \( 3 \cdot \frac{1}{3} - 4 \left(\frac{1}{3}\right)^3 = 1 - \frac{4}{27} = \frac{27}{27} - \frac{4}{27} = \frac{23}{27} \).

3. **Thus, \( 3 \sin^{-1}\left(\frac{1}{3}\right) = \sin^{-1}\left(\frac{23}{27}\right) \)**.

4. **Now, add \( \sin^{-1}\left(\frac{3}{5}\right) \)**:
- We know \( \frac{3}{5} = 0.6 \), and \( \sin^{-1}\left(\frac{3}{5}\right) \) is less than \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \).

5. **Estimate \( B \)**:
- Since \( \frac{23}{27} < 1 \), we can find \( B < \sin^{-1}\left(1\right) = \frac{\pi}{2} \).

### Step 3: Compare \( A \) and \( B \)

- From our evaluations:
- \( A > \frac{2\pi}{3} \)
- \( B < \frac{\pi}{2} \)

Since \( \frac{2\pi}{3} \) is greater than \( \frac{\pi}{2} \), we conclude that:

\[ A > B \]

### Final Answer

Thus, the final result is:

\[ A > B \]