What is the value of `cos46^@ cos 47^@ cos48^@ cos49^@ cos50^@ ..................cos135^@`

Options

A. `-1`
B. 0
C. 1
D. Greater than 1

Answer

To find the value of the expression \( \cos 46^\circ \cos 47^\circ \cos 48^\circ \cos 49^\circ \cos 50^\circ \ldots \cos 135^\circ \), we can follow these steps:

### Step 1: Identify the range of angles
The angles involved in the product are from \( 46^\circ \) to \( 135^\circ \).

### Step 2: Check for any angle that makes the product zero
We know that the cosine function has specific values at certain angles. One critical angle to consider is \( 90^\circ \), where:
\[ \cos 90^\circ = 0 \]

### Step 3: Determine if \( 90^\circ \) is included in the product
Since \( 90^\circ \) falls within the range of angles from \( 46^\circ \) to \( 135^\circ \), it is indeed included in our product.

### Step 4: Conclude the value of the product
Since the product includes \( \cos 90^\circ \), which is equal to 0, the entire product becomes:
\[ \cos 46^\circ \cos 47^\circ \cos 48^\circ \cos 49^\circ \cos 50^\circ \ldots \cos 135^\circ = 0 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0} \]