Consider the following statement :
1. `tan^(-1)1 + tan^(-1) (0.5) = pi//2`
2. `sin^(-1)(1//3) + cos^(-1)(1//3) = pi//2`
Which of the above statement is/are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To solve the given question, we will evaluate each statement one by one.

### Statement 1:
\[ \tan^{-1}(1) + \tan^{-1}(0.5) = \frac{\pi}{2} \]

1. **Evaluate \(\tan^{-1}(1)\)**:
\[ \tan^{-1}(1) = \frac{\pi}{4} \]
This is because the tangent of \(\frac{\pi}{4}\) is 1.

2. **Evaluate \(\tan^{-1}(0.5)\)**:
\[ \tan^{-1}(0.5) \text{ is not a standard angle, but we can denote it as } \theta. \]
Thus, we have:
\[ \tan(\theta) = 0.5 \]

3. **Using the formula for the sum of arctangents**:
The formula for the sum of two arctangents is:
\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \]
where \(ab < 1\).

Here, \(a = 1\) and \(b = 0.5\):
\[ \tan^{-1}(1) + \tan^{-1}(0.5) = \tan^{-1}\left(\frac{1 + 0.5}{1 - (1)(0.5)}\right) = \tan^{-1}\left(\frac{1.5}{0.5}\right) = \tan^{-1}(3) \]

4. **Comparing with \(\frac{\pi}{2}\)**:
Since \(\tan^{-1}(3) \neq \frac{\pi}{2}\), the first statement is **false**.

### Statement 2:
\[ \sin^{-1}\left(\frac{1}{3}\right) + \cos^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{2} \]

1. **Using the identity**:
We know that:
\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]
for any \(x\) in the range \([-1, 1]\).

2. **Substituting \(x = \frac{1}{3}\)**:
Therefore,
\[ \sin^{-1}\left(\frac{1}{3}\right) + \cos^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{2} \]

Thus, the second statement is **true**.

### Conclusion:
- **Statement 1** is false.
- **Statement 2** is true.

### Final Answer:
Only the second statement is correct.

---