If `costheta=(1)/(sqrt(2))`,then find the value of `cos2theta`.

Answer

To find the value of \( \cos 2\theta \) given that \( \cos \theta = \frac{1}{\sqrt{2}} \), we can follow these steps:

### Step 1: Identify the value of \( \theta \)
Given that \( \cos \theta = \frac{1}{\sqrt{2}} \), we recognize that this value corresponds to an angle of \( 45^\circ \) (or \( \frac{\pi}{4} \) radians).

### Step 2: Calculate \( 2\theta \)
Since \( \theta = 45^\circ \), we can find \( 2\theta \) by multiplying \( \theta \) by 2:
\[ 2\theta = 2 \times 45^\circ = 90^\circ \]

### Step 3: Find \( \cos 2\theta \)
Now that we have \( 2\theta = 90^\circ \), we can find \( \cos 2\theta \):
\[ \cos 2\theta = \cos 90^\circ \]
We know that \( \cos 90^\circ = 0 \).

### Final Answer
Thus, the value of \( \cos 2\theta \) is:
\[ \cos 2\theta = 0 \]

---