Which one of the following is true ?

Options

A. `tan x gt 1 , 45^(@) lt x lt 90^(@)`
B. ` sin x gt (1)/(2) , 0^(@) lt x lt 30^(@)`
C. ` cos x gt (1)/(2) , 60^(@) lt x lt 90^(@)`
D. sin x = cos x for some value of ` x , 30^(@) lt x lt 45^(@)`

Answer

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

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

**Solution:**
- We know that \( \tan 45^\circ = 1 \) and \( \tan 90^\circ \) is undefined.
- As \( x \) increases from \( 45^\circ \) to \( 90^\circ \), the value of \( \tan x \) increases from \( 1 \) to infinity.
- Therefore, \( \tan x > 1 \) for \( x \) in the interval \( (45^\circ, 90^\circ) \).

**Conclusion:** Option 1 is correct.

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

**Solution:**
- We know that \( \sin 0^\circ = 0 \) and \( \sin 30^\circ = \frac{1}{2} \).
- As \( x \) increases from \( 0^\circ \) to \( 30^\circ \), \( \sin x \) increases from \( 0 \) to \( \frac{1}{2} \).
- Thus, \( \sin x \) does not exceed \( \frac{1}{2} \) in this interval.

**Conclusion:** Option 2 is incorrect.

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

**Solution:**
- We know that \( \cos 60^\circ = \frac{1}{2} \) and \( \cos 90^\circ = 0 \).
- As \( x \) increases from \( 60^\circ \) to \( 90^\circ \), \( \cos x \) decreases from \( \frac{1}{2} \) to \( 0 \).
- Therefore, \( \cos x \) is equal to \( \frac{1}{2} \) only at \( 60^\circ \) and is less than \( \frac{1}{2} \) for \( x > 60^\circ \).

**Conclusion:** Option 3 is incorrect.

### Step 4: Analyze Option 4
**Statement:** \( \sin x = \cos x \) for some value of \( x \).

**Solution:**
- The equation \( \sin x = \cos x \) holds true when \( x = 45^\circ \) (and also at \( 225^\circ \), but we are considering angles between \( 0^\circ \) and \( 90^\circ \)).
- At \( x = 45^\circ \), both \( \sin 45^\circ \) and \( \cos 45^\circ \) equal \( \frac{1}{\sqrt{2}} \).
- Thus, there is a value of \( x \) (specifically \( 45^\circ \)) for which \( \sin x = \cos x \).

**Conclusion:** Option 4 is correct.

### Final Conclusion
The only correct options are:
- **Option 1:** Correct
- **Option 2:** Incorrect
- **Option 3:** Incorrect
- **Option 4:** Correct

Thus, the true statements are options 1 and 4.

---