If `tan theta=2,`then the find the value of `(2sinthetacostheta)/(cos^(2)theta-sin^(2)theta)`.

Answer

To solve the problem, we need to find the value of \(\frac{2 \sin \theta \cos \theta}{\cos^2 \theta - \sin^2 \theta}\) given that \(\tan \theta = 2\).

### Step-by-Step Solution:

1. **Express \( \sin \theta \) and \( \cos \theta \) in terms of \( \tan \theta \)**:
Since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we can express \(\sin \theta\) and \(\cos \theta\) as:
\[ \sin \theta = 2 \cos \theta \]

2. **Use the Pythagorean identity**:
We know that \(\sin^2 \theta + \cos^2 \theta = 1\). Substituting \(\sin \theta = 2 \cos \theta\) into the identity gives:
\[ (2 \cos \theta)^2 + \cos^2 \theta = 1 \]
\[ 4 \cos^2 \theta + \cos^2 \theta = 1 \]
\[ 5 \cos^2 \theta = 1 \]
\[ \cos^2 \theta = \frac{1}{5} \]

3. **Find \( \sin^2 \theta \)**:
Using \(\sin^2 \theta = 1 - \cos^2 \theta\):
\[ \sin^2 \theta = 1 - \frac{1}{5} = \frac{4}{5} \]

4. **Substitute \( \sin \theta \) and \( \cos \theta \) into the original expression**:
We need to find:
\[ \frac{2 \sin \theta \cos \theta}{\cos^2 \theta - \sin^2 \theta} \]
First, calculate \(2 \sin \theta \cos \theta\):
\[ 2 \sin \theta \cos \theta = 2(2 \cos \theta)(\cos \theta) = 4 \cos^2 \theta \]
Now calculate \(\cos^2 \theta - \sin^2 \theta\):
\[ \cos^2 \theta - \sin^2 \theta = \frac{1}{5} - \frac{4}{5} = -\frac{3}{5} \]

5. **Combine the results**:
Now substitute these values back into the expression:
\[ \frac{4 \cos^2 \theta}{\cos^2 \theta - \sin^2 \theta} = \frac{4 \cdot \frac{1}{5}}{-\frac{3}{5}} = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3} \]

Thus, the final answer is:
\[ \boxed{-\frac{4}{3}} \]