If `costhetacosec23^(@)=1`, the value of `theta` is

Options

A. `23^(@)`
B. `37^(@)`
C. `63^(@)`
D. `67^(@)`

Answer

To solve the equation \( \cos \theta \cdot \csc 23^\circ = 1 \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \cos \theta \cdot \csc 23^\circ = 1 \]

2. **Rewrite \(\csc 23^\circ\) in terms of \(\sin 23^\circ\):**
\[ \csc 23^\circ = \frac{1}{\sin 23^\circ} \]
Therefore, we can rewrite the equation as:
\[ \cos \theta \cdot \frac{1}{\sin 23^\circ} = 1 \]

3. **Multiply both sides by \(\sin 23^\circ\):**
\[ \cos \theta = \sin 23^\circ \]

4. **Use the co-function identity:**
We know that:
\[ \sin(90^\circ - \theta) = \cos \theta \]
Therefore, we can express \(\sin 23^\circ\) as:
\[ \sin 23^\circ = \cos(90^\circ - 23^\circ) = \cos 67^\circ \]

5. **Set the angles equal to each other:**
Since \(\cos \theta = \cos 67^\circ\), we can conclude:
\[ \theta = 67^\circ \]

### Final Answer:
\[ \theta = 67^\circ \]