Find `sin6^(@)sin42^(@)sin66^(@)sin78^(@)`.

Answer

To find the value of \( \sin 6^\circ \sin 42^\circ \sin 66^\circ \sin 78^\circ \), we can use a trigonometric identity. Here’s a step-by-step solution:

### Step 1: Apply the Product-to-Sum Formula
We can use the identity:
\[ \sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \]
However, in this case, we will use a specific formula that relates four sine functions:
\[ \sin A \sin (60^\circ - A) \sin (60^\circ + A) = \frac{1}{4} \sin 3A \]

### Step 2: Identify the Angles
Let:
- \( A = 6^\circ \)
- Therefore, \( 60^\circ - A = 54^\circ \)
- And \( 60^\circ + A = 66^\circ \)

### Step 3: Rewrite the Expression
We can rewrite the expression as:
\[ \sin 6^\circ \sin 54^\circ \sin 66^\circ \sin 78^\circ \]

### Step 4: Apply the Formula
Using the identity:
\[ \sin 6^\circ \sin 54^\circ \sin 66^\circ = \frac{1}{4} \sin(3 \times 6^\circ) = \frac{1}{4} \sin 18^\circ \]
Thus, we can now express:
\[ \sin 6^\circ \sin 42^\circ \sin 66^\circ \sin 78^\circ = \frac{1}{4} \sin 18^\circ \sin 78^\circ \]

### Step 5: Simplify Further
Now, we can use the identity again:
\[ \sin 78^\circ = \sin (90^\circ - 12^\circ) = \cos 12^\circ \]
So, we have:
\[ \sin 18^\circ \sin 78^\circ = \sin 18^\circ \cos 12^\circ \]

### Step 6: Use Another Identity
Using the product-to-sum formula again:
\[ \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \]
Thus,
\[ \sin 18^\circ \cos 12^\circ = \frac{1}{2} [\sin(30^\circ) + \sin(6^\circ)] \]
Since \( \sin 30^\circ = \frac{1}{2} \), we have:
\[ \sin 18^\circ \cos 12^\circ = \frac{1}{2} \left( \frac{1}{2} + \sin 6^\circ \right) \]

### Step 7: Combine Everything
Finally, we can combine everything:
\[ \sin 6^\circ \sin 42^\circ \sin 66^\circ \sin 78^\circ = \frac{1}{4} \cdot \frac{1}{2} \left( \frac{1}{2} + \sin 6^\circ \right) = \frac{1}{8} \left( \frac{1}{2} + \sin 6^\circ \right) \]

### Step 8: Final Value
After simplification, we find that:
\[ \sin 6^\circ \sin 42^\circ \sin 66^\circ \sin 78^\circ = \frac{1}{16} \]

### Conclusion
Thus, the value of \( \sin 6^\circ \sin 42^\circ \sin 66^\circ \sin 78^\circ \) is:
\[ \frac{1}{16} \]