If `tantheta+sintheta=m and tan theta-sin theta=n`, then find the value of `m^(2)-n^(2)`.

Options

A. a) `sqrt(mn)`
B. b) `2sqrt(mn)`
C. c) `3sqrt(mn)`
D. d) `4sqrt(mn)`

Answer

To solve the problem, we start with the given equations:

1. \( m = \tan \theta + \sin \theta \)
2. \( n = \tan \theta - \sin \theta \)

We need to find the value of \( m^2 - n^2 \).

### Step 1: Use the difference of squares formula
We can use the difference of squares formula, which states that \( a^2 - b^2 = (a - b)(a + b) \). Therefore, we can express \( m^2 - n^2 \) as:

\[ m^2 - n^2 = (m - n)(m + n) \]

### Step 2: Calculate \( m - n \)
Now, we calculate \( m - n \):

\[ m - n = (\tan \theta + \sin \theta) - (\tan \theta - \sin \theta) = \tan \theta + \sin \theta - \tan \theta + \sin \theta = 2\sin \theta \]

### Step 3: Calculate \( m + n \)
Next, we calculate \( m + n \):

\[ m + n = (\tan \theta + \sin \theta) + (\tan \theta - \sin \theta) = \tan \theta + \sin \theta + \tan \theta - \sin \theta = 2\tan \theta \]

### Step 4: Substitute back into the difference of squares formula
Now we substitute \( m - n \) and \( m + n \) back into the difference of squares formula:

\[ m^2 - n^2 = (m - n)(m + n) = (2\sin \theta)(2\tan \theta) \]

### Step 5: Simplify the expression
We simplify the expression:

\[ m^2 - n^2 = 4\sin \theta \tan \theta \]

### Step 6: Use the identity for tangent
Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Therefore, we can rewrite:

\[ m^2 - n^2 = 4\sin \theta \cdot \frac{\sin \theta}{\cos \theta} = \frac{4\sin^2 \theta}{\cos \theta} \]

### Final Result
Thus, the final result for \( m^2 - n^2 \) is:

\[ m^2 - n^2 = 4\sin^2 \theta \sec \theta \]