The product `("cos"(x)/(2))*("cos"(x)/(4))*("cos"(x)/(8))* * * * * * * * ("cos"(x)/(256))` is equal to :

Options

A. `(sin x)/(128 "sin"(x)/(256))`
B. `(sin x)/(256 "sin"(x)/(256))`
C. `(sin x)/(128 "sin"(x)/(128))`
D. `(sin x)/(512 "sin"(x)/(512))`

Answer

To solve the product

\[ P = \left(\frac{\cos x}{2}\right) \cdot \left(\frac{\cos x}{4}\right) \cdot \left(\frac{\cos x}{8}\right) \cdots \left(\frac{\cos x}{256}\right), \]

we can follow these steps:

### Step 1: Rewrite the Product
We can express the product in terms of a single cosine function. The product can be written as:

\[ P = \frac{\cos x}{2} \cdot \frac{\cos x}{4} \cdot \frac{\cos x}{8} \cdots \frac{\cos x}{256} = \cos^n x \cdot \frac{1}{2 \cdot 4 \cdot 8 \cdots 256}, \]

where \( n \) is the number of terms in the product.

### Step 2: Count the Number of Terms
The terms are \( \frac{\cos x}{2}, \frac{\cos x}{4}, \frac{\cos x}{8}, \ldots, \frac{\cos x}{256} \). The denominators are powers of 2, specifically \( 2^1, 2^2, 2^3, \ldots, 2^8 \). Thus, there are 8 terms.

### Step 3: Calculate the Denominator
The product of the denominators is:

\[ 2 \cdot 4 \cdot 8 \cdots 256 = 2^1 \cdot 2^2 \cdot 2^3 \cdots 2^8 = 2^{1+2+3+\ldots+8}. \]

The sum \( 1 + 2 + 3 + \ldots + 8 = \frac{8(8+1)}{2} = 36 \). Therefore,

\[ 2^1 \cdot 2^2 \cdot 2^3 \cdots 2^8 = 2^{36}. \]

### Step 4: Express the Product
Now we can express \( P \) as:

\[ P = \frac{\cos^8 x}{2^{36}}. \]

### Step 5: Simplify the Expression
We can write:

\[ P = \frac{\cos^8 x}{2^{36}} = \frac{1}{2^{36}} \cdot \cos^8 x. \]

### Final Answer
Thus, the product is:

\[ P = \frac{\cos^8 x}{2^{36}}. \]