Statement I The maximum value of `sin theta+costheta` is 2.
Statement II The maximum value of `sin theta` is 1 and that of `cos theta` is also 1.

Options

A. A
B. B
C. C
D. D

Answer

To solve the question, we need to analyze both statements provided:

**Statement I:** The maximum value of \( \sin \theta + \cos \theta \) is 2.

**Statement II:** The maximum value of \( \sin \theta \) is 1 and that of \( \cos \theta \) is also 1.

Let's evaluate each statement step by step.

### Step 1: Analyze Statement II
The maximum value of \( \sin \theta \) and \( \cos \theta \).

- The sine function, \( \sin \theta \), has a maximum value of 1. This occurs at \( \theta = \frac{\pi}{2} + 2n\pi \) for any integer \( n \).
- The cosine function, \( \cos \theta \), also has a maximum value of 1. This occurs at \( \theta = 2n\pi \) for any integer \( n \).

**Conclusion for Statement II:** True, since both \( \sin \theta \) and \( \cos \theta \) can reach a maximum value of 1.

### Step 2: Analyze Statement I
Now, we need to find the maximum value of \( \sin \theta + \cos \theta \).

1. **Use the identity:** We can express \( \sin \theta + \cos \theta \) in a different form. We know that:
\[ \sin \theta + \cos \theta = \sqrt{2} \left( \sin \theta \cdot \frac{1}{\sqrt{2}} + \cos \theta \cdot \frac{1}{\sqrt{2}} \right) \]
This can be rewritten using the sine addition formula:
\[ \sin \theta + \cos \theta = \sqrt{2} \sin \left( \theta + \frac{\pi}{4} \right) \]

2. **Determine the maximum value:** The maximum value of \( \sin \left( \theta + \frac{\pi}{4} \right) \) is 1. Therefore:
\[ \text{Maximum value of } \sin \theta + \cos \theta = \sqrt{2} \cdot 1 = \sqrt{2} \]

3. **Conclusion for Statement I:** The maximum value of \( \sin \theta + \cos \theta \) is \( \sqrt{2} \), not 2. Thus, Statement I is false.

### Final Conclusion
- Statement I is **false**.
- Statement II is **true**.