The value of `cos 1^@ cos 2^@ cos 3^@... cos 179^@` is

Options

A. `(1)/(sqrt(2))`
B. 0
C. 1
D. -1

Explanation

To find the value of \( \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 179^\circ \), we can follow these steps:

### Step 1: Identify the range of angles
We are tasked with multiplying the cosine of angles from \( 1^\circ \) to \( 179^\circ \).

### Step 2: Recognize the special angle
Notice that one of the angles in this range is \( 90^\circ \).

### Step 3: Calculate \( \cos 90^\circ \)
We know that:
\[ \cos 90^\circ = 0 \]

### Step 4: Write the product including \( \cos 90^\circ \)
The product can be expressed as:
\[ \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 89^\circ \cdot \cos 90^\circ \cdot \cos 91^\circ \ldots \cos 179^\circ \]

### Step 5: Substitute \( \cos 90^\circ \) into the product
Since \( \cos 90^\circ = 0 \), we can substitute this into our product:
\[ \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 89^\circ \cdot 0 \cdot \cos 91^\circ \ldots \cos 179^\circ \]

### Step 6: Conclude the product
Any product that includes a factor of \( 0 \) will result in \( 0 \):
\[ \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 179^\circ = 0 \]

### Final Answer
Thus, the value of \( \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 179^\circ \) is:
\[ \boxed{0} \]

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