What is the value of `m^2-n^2`, where `m= tan theta + sin theta and n = tan theta - sin theta` ?

Options

A. `sqrt(m/n)`
B. `4sqrt(mn)`
C. `sqrt(mn)`
D. `4sqrt(m/n)`

Answer

To find the value of \( m^2 - n^2 \) where \( m = \tan \theta + \sin \theta \) and \( n = \tan \theta - \sin \theta \), we can follow these steps:

### Step 1: Write down the expressions for \( m \) and \( n \)
Given:
\[ m = \tan \theta + \sin \theta \]
\[ n = \tan \theta - \sin \theta \]

### Step 2: Use the difference of squares formula
We know that:
\[ m^2 - n^2 = (m+n)(m-n) \]
First, we need to calculate \( m+n \) and \( m-n \).

### Step 3: Calculate \( m+n \) and \( m-n \)
\[ m+n = (\tan \theta + \sin \theta) + (\tan \theta - \sin \theta) = 2\tan \theta \]
\[ m-n = (\tan \theta + \sin \theta) - (\tan \theta - \sin \theta) = 2\sin \theta \]

### Step 4: Substitute \( m+n \) and \( m-n \) into the difference of squares formula
Now substituting back into the formula:
\[ m^2 - n^2 = (m+n)(m-n) = (2\tan \theta)(2\sin \theta) = 4\tan \theta \sin \theta \]

### Step 5: Simplify \( 4\tan \theta \sin \theta \)
Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \):
\[ 4\tan \theta \sin \theta = 4 \left(\frac{\sin \theta}{\cos \theta}\right) \sin \theta = 4 \frac{\sin^2 \theta}{\cos \theta} \]

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