`sin 36^(º)sin 72^(º) sin108^(º) sin 144^(º) = `

Options

A. `1//4`
B. `1//16`
C. `3//4`
D. `5//16`

Answer

To solve the expression \( \sin 36^\circ \sin 72^\circ \sin 108^\circ \sin 144^\circ \), we will follow these steps:

### Step 1: Rewrite the sine functions
We know that:
- \( \sin 108^\circ = \sin(180^\circ - 72^\circ) = \sin 72^\circ \)
- \( \sin 144^\circ = \sin(180^\circ - 36^\circ) = \sin 36^\circ \)

Thus, we can rewrite the expression:
\[ \sin 36^\circ \sin 72^\circ \sin 108^\circ \sin 144^\circ = \sin 36^\circ \sin 72^\circ \sin 72^\circ \sin 36^\circ \]

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

### Step 3: Use known values of sine
We need to know the values of \( \sin 36^\circ \) and \( \sin 72^\circ \):
- \( \sin 36^\circ = \frac{\sqrt{5} - 1}{4} \)
- \( \sin 72^\circ = \frac{\sqrt{5} + 1}{4} \)

### Step 4: Substitute the values
Substituting these values into the expression:
\[ (\sin 36^\circ)^2 = \left(\frac{\sqrt{5} - 1}{4}\right)^2 = \frac{(\sqrt{5} - 1)^2}{16} \]
\[ (\sin 72^\circ)^2 = \left(\frac{\sqrt{5} + 1}{4}\right)^2 = \frac{(\sqrt{5} + 1)^2}{16} \]

### Step 5: Calculate the squares
Calculating the squares:
\[ (\sqrt{5} - 1)^2 = 5 - 2\sqrt{5} + 1 = 6 - 2\sqrt{5} \]
\[ (\sqrt{5} + 1)^2 = 5 + 2\sqrt{5} + 1 = 6 + 2\sqrt{5} \]

### Step 6: Multiply the results
Now, we multiply these two results:
\[ (\sin 36^\circ)^2 (\sin 72^\circ)^2 = \frac{(6 - 2\sqrt{5})(6 + 2\sqrt{5})}{16 \cdot 16} \]
Using the difference of squares:
\[ = \frac{6^2 - (2\sqrt{5})^2}{256} = \frac{36 - 20}{256} = \frac{16}{256} = \frac{1}{16} \]

### Step 7: Final answer
Thus, the final value of the expression \( \sin 36^\circ \sin 72^\circ \sin 108^\circ \sin 144^\circ \) is:
\[ \frac{1}{16} \]