`(2sintheta)/(costheta(1+tan^(2)theta))` simplifies to :

Options

A. `costheta`
B. `cos2theta`
C. `sin2 theta`
D. `sintheta`

Answer

To simplify the expression \(\frac{2 \sin \theta}{\cos \theta (1 + \tan^2 \theta)}\), we can follow these steps:

### Step 1: Rewrite the expression
Start with the given expression:
\[ \frac{2 \sin \theta}{\cos \theta (1 + \tan^2 \theta)} \]

### Step 2: Use the identity for \(1 + \tan^2 \theta\)
Recall the trigonometric identity:
\[ 1 + \tan^2 \theta = \sec^2 \theta \]
Thus, we can rewrite the expression as:
\[ \frac{2 \sin \theta}{\cos \theta \sec^2 \theta} \]

### Step 3: Rewrite \(\sec^2 \theta\)
Since \(\sec \theta = \frac{1}{\cos \theta}\), we have:
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \]
Substituting this into our expression gives:
\[ \frac{2 \sin \theta}{\cos \theta \cdot \frac{1}{\cos^2 \theta}} = \frac{2 \sin \theta \cdot \cos^2 \theta}{\cos \theta} \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ \frac{2 \sin \theta \cdot \cos^2 \theta}{\cos \theta} = 2 \sin \theta \cdot \cos \theta \]

### Step 5: Use the double angle identity
Recall the double angle identity:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Thus, we can write:
\[ 2 \sin \theta \cos \theta = \sin 2\theta \]

### Final Result
Therefore, the simplified form of the expression \(\frac{2 \sin \theta}{\cos \theta (1 + \tan^2 \theta)}\) is:
\[ \sin 2\theta \]

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