What is the value of sin `18^@ cos 36^@`?

Options

A. 4
B. 2
C. 1
D. `1/4`

Answer

To find the value of \( \sin 18^\circ \cos 36^\circ \), we can follow these steps:

### Step 1: Identify the values of \( \sin 18^\circ \) and \( \cos 36^\circ \)
From trigonometric identities, we know:
\[ \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \]
\[ \cos 36^\circ = \frac{\sqrt{5} + 1}{4} \]

### Step 2: Multiply \( \sin 18^\circ \) and \( \cos 36^\circ \)
Now we can multiply these two values:
\[ \sin 18^\circ \cos 36^\circ = \left(\frac{\sqrt{5} - 1}{4}\right) \left(\frac{\sqrt{5} + 1}{4}\right) \]

### Step 3: Simplify the multiplication
The multiplication can be simplified as follows:
\[ = \frac{(\sqrt{5} - 1)(\sqrt{5} + 1)}{16} \]
Using the identity \( (a - b)(a + b) = a^2 - b^2 \):
\[ = \frac{(\sqrt{5})^2 - (1)^2}{16} \]
\[ = \frac{5 - 1}{16} \]
\[ = \frac{4}{16} \]

### Step 4: Final simplification
Now we simplify \( \frac{4}{16} \):
\[ = \frac{1}{4} \]

### Conclusion
Thus, the value of \( \sin 18^\circ \cos 36^\circ \) is:
\[ \frac{1}{4} \]