If `cos theta cosec 23^@ = 1`, then 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 1: Rewrite the equation using trigonometric identities
We know that \( \csc 23^\circ = \frac{1}{\sin 23^\circ} \). Therefore, we can rewrite the equation as:
\[ \cos \theta \cdot \frac{1}{\sin 23^\circ} = 1 \]

### Step 2: Isolate \( \cos \theta \)
Multiplying both sides by \( \sin 23^\circ \) gives:
\[ \cos \theta = \sin 23^\circ \]

### Step 3: Use the co-function identity
We can use the identity \( \sin A = \cos(90^\circ - A) \). Therefore, we can write:
\[ \sin 23^\circ = \cos(90^\circ - 23^\circ) = \cos 67^\circ \]

### Step 4: Set the equations equal
Now we have:
\[ \cos \theta = \cos 67^\circ \]

### Step 5: Solve for \( \theta \)
From the equation \( \cos \theta = \cos 67^\circ \), we can conclude that:
\[ \theta = 67^\circ \]

Thus, the value of \( \theta \) is \( 67^\circ \).

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

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