The value of `"cos"(2pi)/(15)."cos"(4pi)/(15)."cos"(8pi)/(15)."cos"(16pi)/(15)` is

Options

A. (a)`-1/(16)`
B. (b)`-1/(8)`
C. (c) `1/(8)`
D. (d)`1/(16)`

Answer

To solve the problem, we need to find the value of the expression:

\[ \cos\left(\frac{2\pi}{15}\right) \cdot \cos\left(\frac{4\pi}{15}\right) \cdot \cos\left(\frac{8\pi}{15}\right) \cdot \cos\left(\frac{16\pi}{15}\right) \]

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

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

\[ = \frac{1}{2 \sin\left(\frac{2\pi}{15}\right)} \cdot 2 \sin\left(\frac{2\pi}{15}\right) \cdot \cos\left(\frac{2\pi}{15}\right) \cdot \cos\left(\frac{4\pi}{15}\right) \cdot \cos\left(\frac{8\pi}{15}\right) \cdot \cos\left(\frac{16\pi}{15}\right) \]

### Step 2: Apply the Formula \(2 \sin \theta \cos \theta = \sin(2\theta)\)

Using the identity \(2 \sin \theta \cos \theta = \sin(2\theta)\):

\[ = \frac{1}{2 \sin\left(\frac{2\pi}{15}\right)} \cdot \sin\left(\frac{4\pi}{15}\right) \cdot \cos\left(\frac{4\pi}{15}\right) \cdot \cos\left(\frac{8\pi}{15}\right) \cdot \cos\left(\frac{16\pi}{15}\right) \]

### Step 3: Multiply and Divide by \(2\)

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

\[ = \frac{1}{4 \sin\left(\frac{2\pi}{15}\right)} \cdot \sin\left(\frac{4\pi}{15}\right) \cdot 2 \sin\left(\frac{4\pi}{15}\right) \cdot \cos\left(\frac{8\pi}{15}\right) \cdot \cos\left(\frac{16\pi}{15}\right) \]

### Step 4: Apply the Formula Again

Using the identity again:

\[ = \frac{1}{4 \sin\left(\frac{2\pi}{15}\right)} \cdot \sin\left(\frac{8\pi}{15}\right) \cdot \cos\left(\frac{8\pi}{15}\right) \cdot \cos\left(\frac{16\pi}{15}\right) \]

### Step 5: Multiply and Divide by \(2\) Again

We repeat the process:

\[ = \frac{1}{8 \sin\left(\frac{2\pi}{15}\right)} \cdot \sin\left(\frac{8\pi}{15}\right) \cdot 2 \sin\left(\frac{8\pi}{15}\right) \cdot \cos\left(\frac{16\pi}{15}\right) \]

### Step 6: Apply the Formula Once More

Using the identity again:

\[ = \frac{1}{8 \sin\left(\frac{2\pi}{15}\right)} \cdot \sin\left(\frac{16\pi}{15}\right) \cdot \cos\left(\frac{16\pi}{15}\right) \]

### Step 7: Multiply and Divide by \(2\) Again

We multiply and divide by \(2\):

\[ = \frac{1}{16 \sin\left(\frac{2\pi}{15}\right)} \cdot \sin\left(\frac{32\pi}{15}\right) \]

### Step 8: Simplify Using the Sine Function Identity

Using the identity \(\sin(2\pi + \theta) = \sin(\theta)\):

\[ \sin\left(\frac{32\pi}{15}\right) = \sin\left(2\pi + \frac{2\pi}{15}\right) = \sin\left(\frac{2\pi}{15}\right) \]

### Step 9: Final Simplification

Now we can simplify:

\[ = \frac{1}{16 \sin\left(\frac{2\pi}{15}\right)} \cdot \sin\left(\frac{2\pi}{15}\right) = \frac{1}{16} \]

Thus, the value of the expression is:

\[ \boxed{\frac{1}{16}} \]