The value of `sin12^(@)sin24^(@)sin48^(@)sin84^(@),` is

Options

A. `cos20^(@)cos40^(@)cos60^(@)cos80^(@)`
B. 1/16
C. `sin20^(@)sin40^(@)sin60^(@)sin80^(@)`
D. `3//15`

Answer

To find the value of \( \sin 12^\circ \sin 24^\circ \sin 48^\circ \sin 84^\circ \), we can use trigonometric identities and properties. Let's break it down step by step.

### Step 1: Rewrite the Expression
We start with:
\[ \sin 12^\circ \sin 24^\circ \sin 48^\circ \sin 84^\circ \]
We can multiply and divide the expression by 4:
\[ = \frac{1}{4} \cdot 4 \sin 12^\circ \sin 24^\circ \sin 48^\circ \sin 84^\circ \]

### Step 2: Pair the Sine Functions
We can pair the sine functions:
\[ = \frac{1}{4} \cdot (2 \sin 12^\circ \sin 48^\circ) \cdot (2 \sin 24^\circ \sin 84^\circ) \]

### Step 3: Use the Product-to-Sum Formulas
Using the identity \( 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \):
1. For \( 2 \sin 12^\circ \sin 48^\circ \):
\[ 2 \sin 12^\circ \sin 48^\circ = \cos(48^\circ - 12^\circ) - \cos(48^\circ + 12^\circ) = \cos 36^\circ - \cos 60^\circ \]

2. For \( 2 \sin 24^\circ \sin 84^\circ \):
\[ 2 \sin 24^\circ \sin 84^\circ = \cos(84^\circ - 24^\circ) - \cos(84^\circ + 24^\circ) = \cos 60^\circ - \cos 108^\circ \]

### Step 4: Substitute Back into the Expression
Substituting these back, we have:
\[ = \frac{1}{4} \left( \cos 36^\circ - \cos 60^\circ \right) \left( \cos 60^\circ - \cos 108^\circ \right) \]

### Step 5: Evaluate the Cosine Values
We know:
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \cos 108^\circ = -\sin 18^\circ = -\frac{\sqrt{5}-1}{4} \) (since \( \cos 108^\circ = \cos(90^\circ + 18^\circ) = -\sin 18^\circ \))
- \( \cos 36^\circ = \frac{\sqrt{5}+1}{4} \)

### Step 6: Substitute and Simplify
Now substituting these values:
\[ = \frac{1}{4} \left( \frac{\sqrt{5}+1}{4} - \frac{1}{2} \right) \left( \frac{1}{2} - \left(-\frac{\sqrt{5}-1}{4}\right) \right) \]

### Step 7: Simplify Each Part
1. Simplifying \( \frac{\sqrt{5}+1}{4} - \frac{1}{2} \):
\[ = \frac{\sqrt{5}+1 - 2}{4} = \frac{\sqrt{5}-1}{4} \]

2. Simplifying \( \frac{1}{2} + \frac{\sqrt{5}-1}{4} \):
\[ = \frac{2}{4} + \frac{\sqrt{5}-1}{4} = \frac{2 + \sqrt{5} - 1}{4} = \frac{\sqrt{5}+1}{4} \]

### Step 8: Combine the Results
Now we combine:
\[ = \frac{1}{4} \cdot \frac{\sqrt{5}-1}{4} \cdot \frac{\sqrt{5}+1}{4} \]
This can be simplified using the difference of squares:
\[ = \frac{1}{4} \cdot \frac{(\sqrt{5})^2 - (1)^2}{16} = \frac{1}{4} \cdot \frac{5 - 1}{16} = \frac{1}{4} \cdot \frac{4}{16} = \frac{1}{16} \]

### Final Result
Thus, the value of \( \sin 12^\circ \sin 24^\circ \sin 48^\circ \sin 84^\circ \) is:
\[ \boxed{\frac{1}{16}} \]