`sin36^(@).sin72^(@).sin108^(@).sin144^(@)=?`

Options

A. `(5)/(16)`
B. `(16)/(5)`
C. `(13)/(5)`
D. `(13)/(16)`

Answer

To solve the expression \( \sin 36^\circ \cdot \sin 72^\circ \cdot \sin 108^\circ \cdot \sin 144^\circ \), we can use some properties of sine and trigonometric identities. Here’s a step-by-step solution:

### Step 1: Rewrite the Sine Functions
We can use the identity \( \sin(180^\circ - x) = \sin x \) to rewrite some of the sine terms:
- \( \sin 108^\circ = \sin(180^\circ - 72^\circ) = \sin 72^\circ \)
- \( \sin 144^\circ = \sin(180^\circ - 36^\circ) = \sin 36^\circ \)

Thus, the expression becomes:
\[ \sin 36^\circ \cdot \sin 72^\circ \cdot \sin 72^\circ \cdot \sin 36^\circ \]

### Step 2: Combine Like Terms
Now we can combine the terms:
\[ (\sin 36^\circ)^2 \cdot (\sin 72^\circ)^2 \]

### Step 3: Use Known Values for Sine
We can use known values for \( \sin 36^\circ \) and \( \sin 72^\circ \):
- \( \sin 36^\circ = \frac{\sqrt{5} - 1}{4} \)
- \( \sin 72^\circ = \frac{\sqrt{5} + 1}{4} \)

Substituting these values into the expression gives:
\[ \left(\frac{\sqrt{5} - 1}{4}\right)^2 \cdot \left(\frac{\sqrt{5} + 1}{4}\right)^2 \]

### Step 4: Simplify the Expression
Now we can simplify:
\[ \frac{(\sqrt{5} - 1)^2}{16} \cdot \frac{(\sqrt{5} + 1)^2}{16} = \frac{(\sqrt{5} - 1)^2 \cdot (\sqrt{5} + 1)^2}{256} \]

### Step 5: Apply the Difference of Squares
Using the difference of squares:
\[ (\sqrt{5} - 1)(\sqrt{5} + 1) = 5 - 1 = 4 \]
Thus, we have:
\[ (\sqrt{5} - 1)^2 \cdot (\sqrt{5} + 1)^2 = (4)^2 = 16 \]

### Step 6: Final Calculation
Now substituting back:
\[ \frac{16}{256} = \frac{1}{16} \]

### Final Answer
Thus, the value of \( \sin 36^\circ \cdot \sin 72^\circ \cdot \sin 108^\circ \cdot \sin 144^\circ \) is:
\[ \frac{1}{16} \]

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