The value of `cos1^(@)cos2^(@)cos3^(@)…cos180^(@)` is

Options

A. 0
B. 1
C. `(sqrt(3))/(2)`
D. `(1)/(2)`

Answer

To find the value of \( \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 180^\circ \), we can follow these steps:

### Step 1: Understand the range of cosine values
The cosine function, \( \cos \theta \), is defined for angles from \( 0^\circ \) to \( 180^\circ \). We will evaluate the product of cosine values from \( 1^\circ \) to \( 180^\circ \).

**Hint:** Remember that cosine is positive in the first quadrant (from \( 0^\circ \) to \( 90^\circ \)) and negative in the second quadrant (from \( 90^\circ \) to \( 180^\circ \)).

### Step 2: Identify the critical angle
Notice that \( \cos 90^\circ = 0 \). This is a crucial point because it will affect the entire product.

**Hint:** Think about how multiplying by zero affects the product.

### Step 3: Analyze the product
The product \( \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 180^\circ \) includes \( \cos 90^\circ \). Since \( \cos 90^\circ = 0 \), this means that the entire product will be zero.

**Hint:** Consider the impact of having a zero term in a multiplication.

### Step 4: Conclude the value
Since one of the terms in the product is zero, the entire product evaluates to zero.

Thus, the value of \( \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 180^\circ \) is:

\[ \boxed{0} \]