Assertion (A) : Principal value of `cos^(-1)(1)` is `pi`
Reason (R ) : Value of `cos 0` is 1

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 question, we need to analyze both the assertion (A) and the reason (R) provided.

### Step 1: Analyze the Assertion (A)
The assertion states that the principal value of \( \cos^{-1}(1) \) is \( \pi \).

**Explanation**:
The function \( \cos^{-1}(x) \) (or arccosine) gives the angle whose cosine is \( x \). The range of the principal value of \( \cos^{-1}(x) \) is from \( 0 \) to \( \pi \).

Now, we need to find \( \cos^{-1}(1) \). We know that:
\[ \cos(0) = 1 \]
Thus, \( \cos^{-1}(1) = 0 \).

Since \( 0 \) is within the range of \( \cos^{-1}(x) \), we conclude that:
\[ \cos^{-1}(1) = 0 \]
Therefore, the assertion that the principal value of \( \cos^{-1}(1) \) is \( \pi \) is **false**.

### Step 2: Analyze the Reason (R)
The reason states that the value of \( \cos(0) \) is \( 1 \).

**Explanation**:
This is a well-known fact in trigonometry. The cosine of \( 0 \) radians is indeed \( 1 \):
\[ \cos(0) = 1 \]
This statement is **true**.

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

Thus, the correct answer is that Assertion (A) is false and Reason (R) is true.

### Final Answer
The correct option is: A is false and R is true.

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