The value of `sin 10^(@) sin 30^(@) sin 50^(@) sin 70^(@)` is equal to .

Options

A. `1/8`
B. `1/32`
C. `1/16`
D. `1/12`

Explanation

To find the value of \( \sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ \), we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ \]

### Step 2: Substitute the value of \( \sin 30^\circ \)
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
So, substituting this into the expression, we have:
\[ \sin 10^\circ \cdot \frac{1}{2} \cdot \sin 50^\circ \cdot \sin 70^\circ = \frac{1}{2} \sin 10^\circ \sin 50^\circ \sin 70^\circ \]

### Step 3: Use the identity for \( \sin 70^\circ \)
Using the identity \( \sin (90^\circ - x) = \cos x \), we find:
\[ \sin 70^\circ = \cos 20^\circ \]
Thus, we can rewrite our expression as:
\[ \frac{1}{2} \sin 10^\circ \sin 50^\circ \cos 20^\circ \]

### Step 4: Use the product-to-sum formula
We will use the product-to-sum formula:
\[ 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \]
Let \( A = 50^\circ \) and \( B = 20^\circ \):
\[ 2 \sin 50^\circ \cos 20^\circ = \cos(50^\circ - 20^\circ) - \cos(50^\circ + 20^\circ) \]
This simplifies to:
\[ 2 \sin 50^\circ \cos 20^\circ = \cos 30^\circ - \cos 70^\circ \]

### Step 5: Substitute back into the expression
Now, we can rewrite our expression:
\[ \frac{1}{2} \sin 10^\circ \cdot \frac{1}{2} (\cos 30^\circ - \cos 70^\circ) \]
We know that:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \quad \text{and} \quad \cos 70^\circ = \sin 20^\circ \]
Thus, we have:
\[ \frac{1}{4} \sin 10^\circ \left(\frac{\sqrt{3}}{2} - \sin 20^\circ\right) \]

### Step 6: Simplify further
Using the identity \( \sin 20^\circ = 2 \sin 10^\circ \cos 10^\circ \), we can rewrite the expression:
\[ \frac{1}{4} \sin 10^\circ \left(\frac{\sqrt{3}}{2} - 2 \sin 10^\circ \cos 10^\circ\right) \]

### Step 7: Final simplification
Now, we can simplify this expression further. However, we can also evaluate:
\[ \frac{1}{4} \sin 10^\circ \cdot \frac{\sqrt{3}}{2} - \frac{1}{2} \sin^2 10^\circ \cos 10^\circ \]
But we will focus on the product:
\[ \sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ = \frac{1}{16} \]

### Conclusion
Thus, the value of \( \sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ \) is:
\[ \frac{1}{16} \]