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

Options

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

Answer

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

### Step 1: Understand the Product
We need to evaluate the product of cosine values from \( \cos 1^\circ \) to \( \cos 179^\circ \).

### Step 2: Identify Key Values
Notice that \( \cos 90^\circ = 0 \). This is a crucial point because multiplying any number by zero results in zero.

### Step 3: Break Down the Product
The product can be expressed as:
\[ \cos 1^\circ \cdot \cos 2^\circ \cdots \cos 89^\circ \cdot \cos 90^\circ \cdot \cos 91^\circ \cdots \cos 179^\circ \]

### Step 4: Recognize the Impact of \( \cos 90^\circ \)
Since \( \cos 90^\circ = 0 \), the entire product becomes:
\[ \cos 1^\circ \cdot \cos 2^\circ \cdots \cos 89^\circ \cdot 0 \cdot \cos 91^\circ \cdots \cos 179^\circ = 0 \]

### Conclusion
Thus, the value of the product \( \cos 1^\circ \cdot \cos 2^\circ \cdots \cos 179^\circ \) is:
\[ \boxed{0} \]