Assertion (A) : Principal value of `sin^(-1)((1)/(sqrt2))` is `(pi)/(4)`
Reason (R ) : Principal value of `cot^(-1)((-1)/(sqrt3))` is `(pi)/(3)`

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is True

Answer

To solve the given question, we need to analyze both the assertion (A) and the reason (R) separately.

### Step 1: Analyze the Assertion (A)
The assertion states that the principal value of \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \) is \( \frac{\pi}{4} \).

1. **Understanding the Range of \( \sin^{-1}(x) \)**:
The range of the function \( \sin^{-1}(x) \) is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

2. **Finding the Value**:
We need to find \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \).
We know that:
\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Therefore, \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \).

**Conclusion for Assertion**: The assertion is true.

### Step 2: Analyze the Reason (R)
The reason states that the principal value of \( \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \) is \( \frac{\pi}{3} \).

1. **Understanding the Principal Value of \( \cot^{-1}(x) \)**:
The principal value of \( \cot^{-1}(x) \) is defined in the range \( (0, \pi) \).

2. **Finding the Value**:
We need to find \( \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \).
We know that:
\[ \cot^{-1}(-x) = \pi - \cot^{-1}(x) \]
Thus,
\[ \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) = \pi - \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) \]
Now, \( \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) \) corresponds to an angle where \( \cot(\theta) = \frac{1}{\sqrt{3}} \), which is \( \frac{\pi}{6} \) (since \( \cot\left(\frac{\pi}{6}\right) = \sqrt{3} \)).
Therefore,
\[ \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \]

**Conclusion for Reason**: The reason is false because \( \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \) is actually \( \frac{5\pi}{6} \), not \( \frac{\pi}{3} \).

### Final Conclusion
- Assertion (A) is true.
- Reason (R) is false.

Thus, the final answer is that option 3 is correct: A is true, but R is false.

---