The product `cos1^(@)cos2^(@)cos3^(@)cos4^(@)......cos100^(@)`. Is equal to

Options

A. `-1`
B. `(1)/(4)`
C. 1
D. 0

Answer

To solve the problem of finding the product \( \cos(1^\circ) \cos(2^\circ) \cos(3^\circ) \ldots \cos(100^\circ) \), we can follow these steps:

### Step 1: Write the product explicitly
We start with the expression:
\[ P = \cos(1^\circ) \cos(2^\circ) \cos(3^\circ) \ldots \cos(100^\circ) \]

### Step 2: Identify a key cosine value
Notice that \( \cos(90^\circ) = 0 \). This is a crucial observation because if any term in a product is zero, the entire product becomes zero.

### Step 3: Include \( \cos(90^\circ) \) in the product
In our product \( P \), we have \( \cos(90^\circ) \) included since it is part of the range from \( 1^\circ \) to \( 100^\circ \).

### Step 4: Conclude the product
Since \( \cos(90^\circ) = 0 \), we can conclude:
\[ P = \cos(1^\circ) \cos(2^\circ) \ldots \cos(90^\circ) \ldots \cos(100^\circ) = 0 \]

### Final Answer
Thus, the product \( \cos(1^\circ) \cos(2^\circ) \cos(3^\circ) \ldots \cos(100^\circ) \) is equal to:
\[ \boxed{0} \]