If `tan theta + sin theta = a and tan theta - sin theta = b," then " a^2 - b^2` is

Options

A. `sqrt(ab)`
B. `4sqrt(ab)`
C. `4ab`
D. `ab`

Answer

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

1. \( \tan \theta + \sin \theta = a \) (Equation 1)
2. \( \tan \theta - \sin \theta = b \) (Equation 2)

We need to find \( a^2 - b^2 \).

### Step 1: Use the difference of squares formula
We can use the identity \( a^2 - b^2 = (a + b)(a - b) \).

### Step 2: Calculate \( a + b \)
Adding Equation 1 and Equation 2:
\[ a + b = (\tan \theta + \sin \theta) + (\tan \theta - \sin \theta) = 2\tan \theta \]

### Step 3: Calculate \( a - b \)
Subtracting Equation 2 from Equation 1:
\[ a - b = (\tan \theta + \sin \theta) - (\tan \theta - \sin \theta) = 2\sin \theta \]

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

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

### Final Result
Thus, we find that:
\[ a^2 - b^2 = 4\tan \theta \sin \theta \]