The value of `cos "" (pi)/( 15) cos "" (2pi )/( 15)cos "" ( 3pi )/( 15)cos"" ( 4pi )/( 15)cos "" ( 5pi )/( 15) cos ""( 6pi)/( 15)cos "" ( 7 pi )/( 15) `is

Options

A. `1//2^(6)`
B. `1//2^(7)`
C. `1//2^(8)`
D. none of these

Answer

To find the value of

\[ \cos\left(\frac{\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{3\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{5\pi}{15}\right) \cos\left(\frac{6\pi}{15}\right) \cos\left(\frac{7\pi}{15}\right), \]

we will follow these steps:

### Step 1: Multiply and Divide by \(2 \sin\left(\frac{\pi}{15}\right)\)

We start by multiplying and dividing the entire expression by \(2 \sin\left(\frac{\pi}{15}\right)\):

\[ \frac{2 \sin\left(\frac{\pi}{15}\right) \cos\left(\frac{\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{3\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{5\pi}{15}\right) \cos\left(\frac{6\pi}{15}\right) \cos\left(\frac{7\pi}{15}\right)}{2 \sin\left(\frac{\pi}{15}\right)}. \]

### Step 2: Apply the Double Angle Identity

Using the identity \(2 \sin A \cos A = \sin(2A)\), we can rewrite the numerator:

\[ = \frac{\sin\left(\frac{2\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{3\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{5\pi}{15}\right) \cos\left(\frac{6\pi}{15}\right) \cos\left(\frac{7\pi}{15}\right)}{2 \sin\left(\frac{\pi}{15}\right)}. \]

### Step 3: Repeat the Process

Now, we will multiply and divide again by \(2\):

\[ = \frac{2 \sin\left(\frac{2\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{3\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{5\pi}{15}\right) \cos\left(\frac{6\pi}{15}\right) \cos\left(\frac{7\pi}{15}\right)}{4 \sin\left(\frac{\pi}{15}\right)}. \]

Using the identity again, we have:

\[ = \frac{\sin\left(\frac{4\pi}{15}\right) \cos\left(\frac{3\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{5\pi}{15}\right) \cos\left(\frac{6\pi}{15}\right) \cos\left(\frac{7\pi}{15}\right)}{4 \sin\left(\frac{\pi}{15}\right)}. \]

### Step 4: Continue the Process

Continuing this process, we will eventually arrive at:

\[ = \frac{\sin\left(\frac{16\pi}{15}\right)}{2^n \sin\left(\frac{\pi}{15}\right)}, \]

where \(n\) is the number of times we multiplied by 2.

### Step 5: Simplify the Expression

Since \(\sin\left(\frac{16\pi}{15}\right) = -\sin\left(\frac{\pi}{15}\right)\):

\[ = \frac{-\sin\left(\frac{\pi}{15}\right)}{2^n \sin\left(\frac{\pi}{15}\right)}. \]

### Step 6: Cancel Out \(\sin\left(\frac{\pi}{15}\right)\)

This simplifies to:

\[ = \frac{-1}{2^n}. \]

### Final Step: Determine the Value of \(n\)

After performing the multiplications and divisions, we find that \(n = 7\). Thus, the final value is:

\[ \frac{-1}{2^7} = -\frac{1}{128}. \]

### Conclusion

The value of

\[ \cos\left(\frac{\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{3\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{5\pi}{15}\right) \cos\left(\frac{6\pi}{15}\right) \cos\left(\frac{7\pi}{15}\right) = \frac{1}{128}. \]