The value of `cos(pi/2^(2)).cos(pi/2^(3))……….cos(pi/2^(10)).sin(pi/2^(10))` is

Options

A. `1/256`
B. `1/2`
C. `1/512`
D. `1/1024`

Explanation

To find the value of the expression \( \cos\left(\frac{\pi}{2^2}\right) \cdot \cos\left(\frac{\pi}{2^3}\right) \cdots \cos\left(\frac{\pi}{2^{10}}\right) \cdot \sin\left(\frac{\pi}{2^{10}}\right) \), we can use a trigonometric identity.

### Step-by-step Solution:

1. **Identify the Pattern**:
We have a product of cosines of the form \( \cos\left(\frac{\pi}{2^2}\right) \cdot \cos\left(\frac{\pi}{2^3}\right) \cdots \cos\left(\frac{\pi}{2^{10}}\right) \) and a sine term \( \sin\left(\frac{\pi}{2^{10}}\right) \).

2. **Use the Trigonometric Identity**:
The identity we will use is:
\[ \cos \theta \cdot \cos 2\theta \cdot \cos 4\theta \cdots \cos(2^{n-1}\theta) = \frac{\sin(2^n \theta)}{2^n \sin \theta} \]
Here, we can let \( \theta = \frac{\pi}{2^{10}} \).

3. **Determine \( n \)**:
In our case, we need to find \( n \) such that the last cosine term is \( \cos\left(\frac{\pi}{2^{10}}\right) \). The terms are \( \cos\left(\frac{\pi}{2^2}\right), \cos\left(\frac{\pi}{2^3}\right), \ldots, \cos\left(\frac{\pi}{2^{10}}\right) \), which gives us \( n = 10 - 2 + 1 = 9 \).

4. **Apply the Identity**:
Now, substituting \( n = 9 \) and \( \theta = \frac{\pi}{2^{10}} \) into the identity, we get:
\[ \cos\left(\frac{\pi}{2^2}\right) \cdot \cos\left(\frac{\pi}{2^3}\right) \cdots \cos\left(\frac{\pi}{2^{10}}\right) = \frac{\sin\left(2^9 \cdot \frac{\pi}{2^{10}}\right)}{2^9 \sin\left(\frac{\pi}{2^{10}}\right)} \]

5. **Simplify the Sine Term**:
Calculate \( 2^9 \cdot \frac{\pi}{2^{10}} = \frac{\pi}{2} \). Thus, we have:
\[ \sin\left(\frac{\pi}{2}\right) = 1 \]
Therefore, the expression simplifies to:
\[ \frac{1}{2^9 \sin\left(\frac{\pi}{2^{10}}\right)} \]

6. **Calculate \( \sin\left(\frac{\pi}{2^{10}}\right) \)**:
Since \( \sin\left(\frac{\pi}{2^{10}}\right) \) is a small angle, we can approximate it as:
\[ \sin\left(\frac{\pi}{2^{10}}\right) \approx \frac{\pi}{2^{10}} \]

7. **Final Calculation**:
Substituting this back, we get:
\[ \frac{1}{2^9 \cdot \frac{\pi}{2^{10}}} = \frac{2^{10}}{2^9 \pi} = \frac{2}{\pi} \]

8. **Final Result**:
Thus, the value of the original expression is:
\[ \frac{2}{\pi} \]