Which of the following is/are true ?

Options

A. `"tan"^(-1)(1)/(3)=(1)/(2)"sin"^(-1)(3)/(5)`
B. `"tan"^(-1)(1)/(3)=(pi)/(4)-cot^(-1)2`
C. `"tan"^(-1)(1)/(3)=(pi)/(4)-(1)/(2)"cos"^(-1)(4)/(5)`
D. `"tan"^(-1)(1)/(3)=(pi)/(2)-cot^(-1)3`

Explanation

To determine which of the statements regarding the inverse trigonometric functions is true, we will analyze each statement step by step.

### Step 1: Evaluate Statement A
**Statement A:** \( \tan^{-1}\left(\frac{1}{3}\right) = \frac{1}{2} \sin^{-1}\left(\frac{3}{5}\right) \)

1. Start with the left side: \( \tan^{-1}\left(\frac{1}{3}\right) \).
2. We need to find \( \sin^{-1}\left(\frac{3}{5}\right) \).
- In a right triangle, if the opposite side is 3 and the hypotenuse is 5, we can find the adjacent side using the Pythagorean theorem:
\[ \text{adjacent} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \]
- Therefore, \( \tan(\theta) = \frac{3}{4} \) implies \( \theta = \tan^{-1}\left(\frac{3}{4}\right) \).
3. Now, using the double angle identity for tangent:
\[ 2\tan^{-1}\left(\frac{1}{3}\right) = \tan^{-1}\left(\frac{2 \cdot \frac{1}{3}}{1 - \left(\frac{1}{3}\right)^2}\right) = \tan^{-1}\left(\frac{\frac{2}{3}}{\frac{8}{9}}\right) = \tan^{-1}\left(\frac{2 \cdot 9}{3 \cdot 8}\right) = \tan^{-1}\left(\frac{3}{4}\right) \]
4. Thus, \( \tan^{-1}\left(\frac{1}{3}\right) = \frac{1}{2} \sin^{-1}\left(\frac{3}{5}\right) \) is true.

### Step 2: Evaluate Statement B
**Statement B:** \( \tan^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{4} - \cot^{-1}\left(\frac{1}{2}\right) \)

1. Recall that \( \cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right) \).
2. Thus, \( \cot^{-1}\left(\frac{1}{2}\right) = \tan^{-1}(2) \).
3. Therefore, we can rewrite the statement as:
\[ \tan^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{4} - \tan^{-1}(2) \]
4. Using the identity:
\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \]
where \( a = \frac{1}{3} \) and \( b = 2 \):
\[ \tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}(2) = \tan^{-1}\left(\frac{\frac{1}{3} + 2}{1 - \frac{1}{3} \cdot 2}\right) = \tan^{-1}\left(\frac{\frac{1}{3} + \frac{6}{3}}{1 - \frac{2}{3}}\right) = \tan^{-1}\left(\frac{\frac{7}{3}}{\frac{1}{3}}\right) = \tan^{-1}(7) \]
5. Hence, the statement is true.

### Step 3: Evaluate Statement C
**Statement C:** \( \tan^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{4} - \frac{1}{2} \cos^{-1}\left(\frac{4}{5}\right) \)

1. We know that \( \cos^{-1}(x) + \sin^{-1}(x) = \frac{\pi}{2} \).
2. Thus, \( \cos^{-1}\left(\frac{4}{5}\right) = \frac{\pi}{2} - \sin^{-1}\left(\frac{3}{5}\right) \).
3. Therefore, we can rewrite:
\[ \frac{1}{2} \cos^{-1}\left(\frac{4}{5}\right) = \frac{1}{2} \left(\frac{\pi}{2} - \sin^{-1}\left(\frac{3}{5}\right)\right) \]
4. This leads to:
\[ \tan^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{4} - \frac{1}{4}\pi + \frac{1}{2} \sin^{-1}\left(\frac{3}{5}\right) \]
5. Therefore, this statement is also true.

### Step 4: Evaluate Statement D
**Statement D:** \( \tan^{-1}\left(-\frac{1}{3}\right) = \frac{\pi}{2} - \cot^{-1}(3) \)

1. We know that \( \tan^{-1}(-x) = -\tan^{-1}(x) \).
2. Thus, \( \tan^{-1}\left(-\frac{1}{3}\right) = -\tan^{-1}\left(\frac{1}{3}\right) \).
3. Also, \( \cot^{-1}(x) = \frac{\pi}{2} - \tan^{-1}(x) \).
4. Therefore, \( \cot^{-1}(3) = \frac{\pi}{2} - \tan^{-1}(3) \).
5. This means that \( \frac{\pi}{2} - \cot^{-1}(3) = \tan^{-1}(3) \), which does not equal \( -\tan^{-1}\left(\frac{1}{3}\right) \).

### Conclusion
- **True Statements:** A, B, C
- **False Statement:** D