The value of `sin10^(@). sin30^(@).sin50^(@). Sin70^(@)` is

Options

A. `(1)/(36)`
B. `(1)/(32)`
C. `(1)/(18)`
D. `(1)/(16)`

Answer

To find the value of \( \sin 10^\circ \cdot \sin 30^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ \), we can use some trigonometric identities and properties.

### Step-by-Step Solution:

1. **Identify the angles**: We have the angles \( 10^\circ, 30^\circ, 50^\circ, \) and \( 70^\circ \).

2. **Use the identity for sine**: Notice that \( \sin 70^\circ = \sin(90^\circ - 20^\circ) = \cos 20^\circ \) and \( \sin 50^\circ = \sin(90^\circ - 40^\circ) = \cos 40^\circ \). However, we will keep them as they are for now.

3. **Group the sine terms**: We can group \( \sin 10^\circ \) with \( \sin 70^\circ \) and \( \sin 30^\circ \) with \( \sin 50^\circ \):
\[ \sin 10^\circ \cdot \sin 70^\circ \cdot \sin 30^\circ \cdot \sin 50^\circ \]

4. **Use the sine product identity**: We can use the identity:
\[ \sin A \cdot \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \]
Specifically, we can use the identity:
\[ \sin x \cdot \sin(60^\circ - x) \cdot \sin(60^\circ + x) = \frac{1}{4} \sin(3x) \]
where \( x = 10^\circ \).

5. **Apply the identity**: Here, we can rewrite:
\[ \sin 10^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ = \sin 10^\circ \cdot \sin(60^\circ - 10^\circ) \cdot \sin(60^\circ + 10^\circ) \]
So, we have:
\[ \sin 10^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ = \frac{1}{4} \sin(30^\circ) \]

6. **Calculate \( \sin 30^\circ \)**: We know that \( \sin 30^\circ = \frac{1}{2} \).

7. **Combine the results**: Therefore:
\[ \sin 10^\circ \cdot \sin 30^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \]

8. **Final calculation**: Now, we multiply this by \( \sin 30^\circ \):
\[ \frac{1}{8} \cdot \frac{1}{2} = \frac{1}{16} \]

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