What is the value of `cos36^@` .

Options

A. `(sqrt(5)+1)/4`
.
B. `(sqrt(5)-1)/4`
.
C. `(sqrt(5)+1)/2`
.
D. None of these

Explanation

To find the value of \( \cos 36^\circ \), we will use trigonometric identities and known values of sine and cosine.

### Step-by-Step Solution:

1. **Use the double angle identity for cosine**:
\[ \cos(2\theta) = 1 - 2\sin^2(\theta) \]
Here, let \( \theta = 18^\circ \). Therefore, \( \cos(36^\circ) = \cos(2 \times 18^\circ) \).

2. **Substitute into the identity**:
\[ \cos(36^\circ) = 1 - 2\sin^2(18^\circ) \]

3. **Use the known value of \( \sin(18^\circ) \)**:
The value of \( \sin(18^\circ) \) is known to be:
\[ \sin(18^\circ) = \frac{\sqrt{5} - 1}{4} \]

4. **Substitute \( \sin(18^\circ) \) into the equation**:
\[ \cos(36^\circ) = 1 - 2\left(\frac{\sqrt{5} - 1}{4}\right)^2 \]

5. **Calculate \( \left(\frac{\sqrt{5} - 1}{4}\right)^2 \)**:
\[ \left(\frac{\sqrt{5} - 1}{4}\right)^2 = \frac{(\sqrt{5} - 1)^2}{16} = \frac{5 - 2\sqrt{5} + 1}{16} = \frac{6 - 2\sqrt{5}}{16} = \frac{3 - \sqrt{5}}{8} \]

6. **Substitute back into the equation for \( \cos(36^\circ) \)**:
\[ \cos(36^\circ) = 1 - 2\left(\frac{3 - \sqrt{5}}{8}\right) \]
\[ = 1 - \frac{6 - 2\sqrt{5}}{8} \]

7. **Simplify the expression**:
\[ = 1 - \frac{6}{8} + \frac{2\sqrt{5}}{8} \]
\[ = 1 - \frac{3}{4} + \frac{\sqrt{5}}{4} \]
\[ = \frac{4}{4} - \frac{3}{4} + \frac{\sqrt{5}}{4} \]
\[ = \frac{1 + \sqrt{5}}{4} \]

8. **Final result**:
\[ \cos(36^\circ) = \frac{\sqrt{5} + 1}{4} \]

### Conclusion:
Thus, the value of \( \cos 36^\circ \) is \( \frac{\sqrt{5} + 1}{4} \).