Statement (A) : `sin1^(@)ltcos1^(@)`
Reason (R ) : `sinthetaltcostheta` when `0^(@)ltthetalt90^(@)`then

Options

A. Both A and R are correct and R is the correct explanation of A .
B. Both A and R are correct but R is not a correct explanation of A .
C. A is correct, R is incorrect .
D. A is incorrect, R is correct .

Answer

To solve the problem, we need to analyze both the statement (A) and the reason (R) given in the question.

### Step 1: Analyze Statement (A)
Statement (A) claims that \( \sin(1^\circ) < \cos(1^\circ) \).

To verify this, we can calculate the values of \( \sin(1^\circ) \) and \( \cos(1^\circ) \):

- \( \sin(1^\circ) \) is approximately 0.017452.
- \( \cos(1^\circ) \) is approximately 0.999847.

Since \( 0.017452 < 0.999847 \), we can conclude that \( \sin(1^\circ) < \cos(1^\circ) \).

**Conclusion for Step 1**: Statement (A) is true.

### Step 2: Analyze Reason (R)
Reason (R) states that \( \sin(\theta) < \cos(\theta) \) when \( 0^\circ < \theta < 90^\circ \).

To understand this, we need to consider the behavior of the sine and cosine functions in the first quadrant (from \( 0^\circ \) to \( 90^\circ \)):

- At \( \theta = 0^\circ \), \( \sin(0^\circ) = 0 \) and \( \cos(0^\circ) = 1 \), so \( \sin(0^\circ) < \cos(0^\circ) \).
- At \( \theta = 45^\circ \), \( \sin(45^\circ) = \cos(45^\circ) = \frac{1}{\sqrt{2}} \), so \( \sin(45^\circ) = \cos(45^\circ) \).
- For \( 0^\circ < \theta < 45^\circ \), \( \sin(\theta) < \cos(\theta) \).
- For \( 45^\circ < \theta < 90^\circ \), \( \sin(\theta) > \cos(\theta) \).

Thus, the statement \( \sin(\theta) < \cos(\theta) \) is only true for \( 0^\circ < \theta < 45^\circ \). Therefore, the reason (R) is not universally true for the entire interval \( 0^\circ < \theta < 90^\circ \).

**Conclusion for Step 2**: Reason (R) is false.

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

Thus, the correct answer is that statement (A) is correct, but reason (R) is incorrect.