If `y=2sin^2theta+tantheta` then `(dy)/(dtheta)` will be

Options

A. `4sinthetacostheta+secthetatantheta`
B. `2sin2theta+sec^2theta`
C. `4sintheta+sec^2theta`
D. `2cos^2theta+sec^2theta`

Explanation

To find \(\frac{dy}{d\theta}\) for the function \(y = 2\sin^2\theta + \tan\theta\), we will differentiate the expression with respect to \(\theta\). Let's go through the steps:

### Step 1: Write down the function
We start with the function:
\[ y = 2\sin^2\theta + \tan\theta \]

### Step 2: Differentiate \(y\) with respect to \(\theta\)
To differentiate \(y\), we will use the chain rule and the derivatives of the trigonometric functions.

1. The derivative of \(2\sin^2\theta\):
- Using the chain rule, we have:
\[ \frac{d}{d\theta}(2\sin^2\theta) = 2 \cdot 2\sin\theta \cdot \frac{d}{d\theta}(\sin\theta) = 2 \cdot 2\sin\theta \cdot \cos\theta = 4\sin\theta\cos\theta \]
- This can be simplified using the double angle identity:
\[ 4\sin\theta\cos\theta = 2\sin(2\theta) \]

2. The derivative of \(\tan\theta\):
- The derivative of \(\tan\theta\) is:
\[ \frac{d}{d\theta}(\tan\theta) = \sec^2\theta \]

### Step 3: Combine the derivatives
Now we combine the derivatives we found:
\[ \frac{dy}{d\theta} = 2\sin(2\theta) + \sec^2\theta \]

### Final Result
Thus, the derivative of \(y\) with respect to \(\theta\) is:
\[ \frac{dy}{d\theta} = 2\sin(2\theta) + \sec^2\theta \]