Which of the following is true ?

Options

A. `tanxgt1, 45^(@)ltxlt90^(@)`
B. `sinxgt(1)/(2),0^(@)ltxlt30^(@)`
C. `cosxgt(1)/(2),60^(@)ltxlt90^(@)`
D. For some value of `x` in `30^(@)ltxlt45^(@),sinx=cosx`

Answer

To determine which of the following statements is true, we will analyze each option step by step.

### Step 1: Analyze Option A
**Statement:** \(10x > 1\) if \(x\) is between \(45^\circ\) and \(90^\circ\).

- **Evaluation:**
- If \(x = 45^\circ\), then \(10 \times 45 = 450\).
- If \(x = 90^\circ\), then \(10 \times 90 = 900\).
- Since \(x\) is between \(45^\circ\) and \(90^\circ\), \(10x\) will take values from \(450\) to \(900\).
- Therefore, \(10x\) is always greater than \(1\).

**Conclusion:** Option A is true.

### Step 2: Analyze Option B
**Statement:** \(\sin x > \frac{1}{2}\) if \(x\) is between \(0^\circ\) and \(30^\circ\).

- **Evaluation:**
- At \(x = 0^\circ\), \(\sin 0 = 0\).
- At \(x = 30^\circ\), \(\sin 30 = \frac{1}{2}\).
- For values of \(x\) between \(0^\circ\) and \(30^\circ\), \(\sin x\) increases from \(0\) to \(\frac{1}{2}\).
- Therefore, \(\sin x\) is not greater than \(\frac{1}{2}\) for all values in this range.

**Conclusion:** Option B is false.

### Step 3: Analyze Option C
**Statement:** \(\cos x > \frac{1}{2}\) if \(x\) is between \(60^\circ\) and \(90^\circ\).

- **Evaluation:**
- At \(x = 60^\circ\), \(\cos 60 = \frac{1}{2}\).
- At \(x = 90^\circ\), \(\cos 90 = 0\).
- For values of \(x\) between \(60^\circ\) and \(90^\circ\), \(\cos x\) decreases from \(\frac{1}{2}\) to \(0\).
- Therefore, \(\cos x\) is not greater than \(\frac{1}{2}\) for all values in this range.

**Conclusion:** Option C is false.

### Step 4: Analyze Option D
**Statement:** For some value of \(x\) in \(30^\circ\) and \(45^\circ\), \(\sin x = \cos x\).

- **Evaluation:**
- At \(x = 30^\circ\), \(\sin 30 = \frac{1}{2}\) and \(\cos 30 = \frac{\sqrt{3}}{2}\).
- At \(x = 45^\circ\), \(\sin 45 = \frac{1}{\sqrt{2}}\) and \(\cos 45 = \frac{1}{\sqrt{2}}\).
- The only point where \(\sin x = \cos x\) is at \(x = 45^\circ\).
- Therefore, there exists a value of \(x\) in this range where \(\sin x = \cos x\).

**Conclusion:** Option D is true.

### Final Conclusion
After analyzing all the options:
- Option A is true.
- Option B is false.
- Option C is false.
- Option D is true.

The correct answer is that **Option A is true**.

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