What is `(2 tan theta)/(1+tan^2theta)` equals to ?

Options

A. cos2`theta`
B. tan2`theta`
C. sin2`theta`
D. cosec2`theta`

Answer

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

### Step 1: Rewrite \(\tan \theta\)
We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Therefore, we can rewrite the expression as:
\[ \frac{2 \tan \theta}{1 + \tan^2 \theta} = \frac{2 \cdot \frac{\sin \theta}{\cos \theta}}{1 + \left(\frac{\sin \theta}{\cos \theta}\right)^2} \]

### Step 2: Simplify the denominator
The denominator can be simplified:
\[ 1 + \tan^2 \theta = 1 + \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} \]
Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can simplify this to:
\[ 1 + \tan^2 \theta = \frac{1}{\cos^2 \theta} \]

### Step 3: Substitute back into the expression
Now substituting back into our expression gives:
\[ \frac{2 \cdot \frac{\sin \theta}{\cos \theta}}{\frac{1}{\cos^2 \theta}} = 2 \sin \theta \cdot \cos^2 \theta \]

### Step 4: Use the double angle identity
We know from trigonometric identities that:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Thus, we can express \(2 \sin \theta \cos \theta\) as \(\sin 2\theta\). Therefore:
\[ 2 \sin \theta \cos^2 \theta = \sin 2\theta \cdot \cos \theta \]

### Step 5: Final simplification
Since we need to express the original expression in terms of \(\sin 2\theta\), we can conclude:
\[ \frac{2 \tan \theta}{1 + \tan^2 \theta} = \sin 2\theta \]

### Final Answer
Thus, the expression \(\frac{2 \tan \theta}{1 + \tan^2 \theta}\) is equal to \(\sin 2\theta\).