If cosec2 x - 2 = 0, then the value of `x(0 lt x lt 90^(@))` is:

Options

A. `15^(@)`
B. `60^(@)`
C. `30^(@)`
D. `45^(@)`

Answer

To solve the equation \( \csc 2x - 2 = 0 \), we will follow these steps:

### Step 1: Rearrange the equation
Start with the given equation:
\[ \csc 2x - 2 = 0 \]
Add 2 to both sides:
\[ \csc 2x = 2 \]

### Step 2: Convert cosecant to sine
Recall that \( \csc \theta = \frac{1}{\sin \theta} \). Therefore, we can rewrite the equation as:
\[ \frac{1}{\sin 2x} = 2 \]

### Step 3: Cross-multiply
To eliminate the fraction, cross-multiply:
\[ 1 = 2 \sin 2x \]

### Step 4: Solve for sine
Now, divide both sides by 2:
\[ \sin 2x = \frac{1}{2} \]

### Step 5: Find the angle
The sine of an angle is \( \frac{1}{2} \) at \( 30^\circ \) and \( 150^\circ \) in the range of \( 0^\circ \) to \( 360^\circ \). However, since we are looking for \( 2x \), we will focus on:
\[ 2x = 30^\circ \quad \text{(since \( 0 < x < 90^\circ \))} \]

### Step 6: Solve for \( x \)
Now, divide both sides by 2 to find \( x \):
\[ x = \frac{30^\circ}{2} = 15^\circ \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{15^\circ} \]

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