If `0^@ lt theta lt 90^@` , find the value of `theta` satisfying 2 sin `theta`=cosec `theta`

Answer

To solve the equation \( 2 \sin \theta = \csc \theta \) for \( 0^\circ < \theta < 90^\circ \), we can follow these steps:

### Step 1: Rewrite the equation
We know that \( \csc \theta = \frac{1}{\sin \theta} \). Therefore, we can rewrite the equation as:
\[ 2 \sin \theta = \frac{1}{\sin \theta} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ 2 \sin^2 \theta = 1 \]

### Step 3: Rearrange the equation
Now, we can rearrange this equation to isolate \( \sin^2 \theta \):
\[ \sin^2 \theta = \frac{1}{2} \]

### Step 4: Take the square root
Taking the square root of both sides, we get:
\[ \sin \theta = \frac{1}{\sqrt{2}} \quad \text{or} \quad \sin \theta = -\frac{1}{\sqrt{2}} \]
Since \( \theta \) is between \( 0^\circ \) and \( 90^\circ \), we discard the negative solution:
\[ \sin \theta = \frac{1}{\sqrt{2}} \]

### Step 5: Find the angle
The value of \( \theta \) for which \( \sin \theta = \frac{1}{\sqrt{2}} \) is:
\[ \theta = 45^\circ \]

### Conclusion
Thus, the value of \( \theta \) satisfying the equation \( 2 \sin \theta = \csc \theta \) in the interval \( 0^\circ < \theta < 90^\circ \) is:
\[ \theta = 45^\circ \]

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