The value of 16 sin `144^@ sin 108^@ sin 72^@ sin 36^@` is equal to

Options

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

Answer

To solve the expression \( 16 \sin 144^\circ \sin 108^\circ \sin 72^\circ \sin 36^\circ \), we can follow these steps:

### Step 1: Simplify the Sine Functions
We can use the identity \( \sin(180^\circ - \theta) = \sin \theta \) to rewrite the sine functions:
- \( \sin 144^\circ = \sin(180^\circ - 36^\circ) = \sin 36^\circ \)
- \( \sin 108^\circ = \sin(180^\circ - 72^\circ) = \sin 72^\circ \)

### Step 2: Substitute the Values
Now we can substitute these values back into the expression:
\[ 16 \sin 144^\circ \sin 108^\circ \sin 72^\circ \sin 36^\circ = 16 \sin 36^\circ \sin 72^\circ \sin 72^\circ \sin 36^\circ \]
This can be simplified to:
\[ 16 \sin^2 36^\circ \sin^2 72^\circ \]

### Step 3: Use Known Values
We know that:
- \( \sin 36^\circ = \frac{\sqrt{5} - 1}{4} \)
- \( \sin 72^\circ = \frac{\sqrt{5} + 1}{4} \)

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

### Step 5: Calculate the Squares
Calculating the squares:
\[ = 16 \cdot \frac{(\sqrt{5} - 1)^2}{16} \cdot \frac{(\sqrt{5} + 1)^2}{16} \]
\[ = \frac{16}{16} \cdot \frac{(\sqrt{5} - 1)^2 (\sqrt{5} + 1)^2}{16} \]
\[ = \frac{(\sqrt{5} - 1)^2 (\sqrt{5} + 1)^2}{1} \]

### Step 6: Use the Identity
Using the identity \( (a - b)(a + b) = a^2 - b^2 \):
\[ = (\sqrt{5})^4 - (1)^4 = 25 - 1 = 24 \]

### Step 7: Final Calculation
Thus, the final value is:
\[ \frac{24}{16} = \frac{3}{2} \]

### Conclusion
The value of \( 16 \sin 144^\circ \sin 108^\circ \sin 72^\circ \sin 36^\circ \) is equal to \( 3 \).