If `y=2 sin^(2) theta + tan theta` then `(dy)/(d theta)` will be-

Options

A. `4 sin theta cos theta + sec theta tan theta`
B. `2 sin 2 theta + sec^(2) theta`
C. `4 sin theta + sec^(2) theta`
D. `2 cos^(2) theta +sec^(2) theta`

Explanation

To find \(\frac{dy}{d\theta}\) for the function \(y = 2 \sin^2 \theta + \tan \theta\), we will differentiate each term with respect to \(\theta\).

### Step-by-step Solution:

1. **Identify the function**:
\[ y = 2 \sin^2 \theta + \tan \theta \]

2. **Differentiate the first term \(2 \sin^2 \theta\)**:
- We use the chain rule for differentiation here. The derivative of \(\sin^2 \theta\) is:
\[ \frac{d}{d\theta}(\sin^2 \theta) = 2 \sin \theta \cdot \frac{d}{d\theta}(\sin \theta) = 2 \sin \theta \cdot \cos \theta \]
- Therefore, the derivative of \(2 \sin^2 \theta\) is:
\[ \frac{d}{d\theta}(2 \sin^2 \theta) = 2 \cdot 2 \sin \theta \cos \theta = 4 \sin \theta \cos \theta \]

3. **Differentiate the second term \(\tan \theta\)**:
- The derivative of \(\tan \theta\) is:
\[ \frac{d}{d\theta}(\tan \theta) = \sec^2 \theta \]

4. **Combine the derivatives**:
- Now, we can combine the derivatives of both terms:
\[ \frac{dy}{d\theta} = 4 \sin \theta \cos \theta + \sec^2 \theta \]

5. **Use the double angle identity**:
- We know that \(4 \sin \theta \cos \theta\) can be rewritten using the double angle identity:
\[ 4 \sin \theta \cos \theta = 2 \sin(2\theta) \]
- Therefore, we can write:
\[ \frac{dy}{d\theta} = 2 \sin(2\theta) + \sec^2 \theta \]

### Final Answer:
\[ \frac{dy}{d\theta} = 2 \sin(2\theta) + \sec^2 \theta \]