If `sec15theta=cosec15theta(0^(@)ltthetalt10^(@))` then value of `theta` is

Options

A. `9^(@)`
B. `5^(@)`
C. `8^(@)`
D. `3^(@)`

Answer

To solve the equation \( \sec(15\theta) = \csc(15\theta) \) for \( 0^\circ < \theta < 10^\circ \), we can follow these steps:

### Step 1: Rewrite the equation
We know that:
\[ \sec(15\theta) = \frac{1}{\cos(15\theta)} \quad \text{and} \quad \csc(15\theta) = \frac{1}{\sin(15\theta)} \]
Thus, we can rewrite the equation as:
\[ \frac{1}{\cos(15\theta)} = \frac{1}{\sin(15\theta)} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ \sin(15\theta) = \cos(15\theta) \]

### Step 3: Use the identity
The equation \( \sin(15\theta) = \cos(15\theta) \) holds true when:
\[ 15\theta = 45^\circ + n \cdot 180^\circ \quad \text{for any integer } n \]
However, since we are looking for \( \theta \) in the range \( 0^\circ < \theta < 10^\circ \), we will consider only the principal solution:
\[ 15\theta = 45^\circ \]

### Step 4: Solve for \( \theta \)
Now, we can solve for \( \theta \):
\[ \theta = \frac{45^\circ}{15} = 3^\circ \]

### Step 5: Verify the range
Since \( 3^\circ \) is indeed within the range \( 0^\circ < \theta < 10^\circ \), it is a valid solution.

### Final Answer
Thus, the value of \( \theta \) is:
\[ \theta = 3^\circ \]

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