If `0ltthetaltphilt90^(@)` then which one of the following is true ?

Options

A. `(sintheta+costheta)^(2)gt2`
B. `(sin^(2)theta+cos^(2)phi)le2`
C. `(sin^(2)theta+cos^(2)phi)lt2`
D. `(sin^(2)theta+cos^(2)phi)gt2`

Answer

To solve the question, we need to analyze the given conditions and options. The question states that if \(0 < \theta < \phi < 90^\circ\), we need to determine which of the following statements is true.

### Step-by-Step Solution:

1. **Understanding the Trigonometric Values**:
- We know the values of sine and cosine for specific angles:
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
- \(\sin 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\cos 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
- \(\cos 60^\circ = \frac{1}{2}\)

2. **Evaluating Each Option**:
- **Option A**: \(\sin^2 \theta + \cos^2 \theta < 2\)
- For any angle \(\theta\), \(\sin^2 \theta + \cos^2 \theta = 1\). Thus, this statement is true, but it does not provide a strict inequality since \(1 < 2\) is always true.

- **Option B**: \(\sin^2 \theta + \cos^2 \phi < 2\)
- Using \(\theta = 30^\circ\) and \(\phi = 45^\circ\):
\[ \sin^2 30^\circ + \cos^2 45^\circ = \left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4} < 2 \]
- This statement is also true, but it does not provide a strict inequality.

- **Option C**: \(\sin^2 \theta + \cos^2 \theta < 2\)
- As previously established, \(\sin^2 \theta + \cos^2 \theta = 1\), thus this statement is true.

3. **Conclusion**:
- All options appear to be true under the given conditions, but since the question asks for which one is true, we can conclude that:
- The most relevant option that holds true for all values of \(\theta\) and \(\phi\) in the specified range is **Option A**: \(\sin^2 \theta + \cos^2 \theta < 2\).

### Final Answer:
**Option A**: \(\sin^2 \theta + \cos^2 \theta < 2\) is true.