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

Answer

To solve the problem where \( \tan \theta = 2 \) and we need to find the value of \( \frac{2 \sin \theta \cos \theta}{\cos^2 \theta - \sin^2 \theta} \), we can follow these steps:

### Step 1: Use the identity for \( \tan \theta \)
Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can express \( \sin \theta \) in terms of \( \cos \theta \):
\[ \sin \theta = 2 \cos \theta \]

### Step 2: Substitute \( \sin \theta \) into the expression
Now substitute \( \sin \theta \) into the expression \( \frac{2 \sin \theta \cos \theta}{\cos^2 \theta - \sin^2 \theta} \):
\[ \frac{2(2 \cos \theta) \cos \theta}{\cos^2 \theta - (2 \cos \theta)^2} \]
This simplifies to:
\[ \frac{4 \cos^2 \theta}{\cos^2 \theta - 4 \cos^2 \theta} \]

### Step 3: Simplify the denominator
Now simplify the denominator:
\[ \cos^2 \theta - 4 \cos^2 \theta = -3 \cos^2 \theta \]
So the expression now looks like:
\[ \frac{4 \cos^2 \theta}{-3 \cos^2 \theta} \]

### Step 4: Cancel out \( \cos^2 \theta \)
Assuming \( \cos^2 \theta \neq 0 \), we can cancel \( \cos^2 \theta \) from the numerator and denominator:
\[ \frac{4}{-3} = -\frac{4}{3} \]

### Final Answer
Thus, the value of \( \frac{2 \sin \theta \cos \theta}{\cos^2 \theta - \sin^2 \theta} \) is:
\[ -\frac{4}{3} \]