Question

If \( \tan \theta + \sin \theta = m \) and \( \tan \theta - \sin \theta = n \), then the value of \( m^2 - n^2 \) is:

• A. \( \sqrt{mn} \)
• B. \( 2\sqrt{mn} \)
• C. \( 4mn \)
• D. \( \sqrt{mn} \)

Hint:

When dealing with expressions involving sums and differences of the same terms, use the identity \( a^2 - b^2 = (a + b)(a - b) \) to simplify.

Answer 1

The Correct Option is C

We are given that: \[ \tan \theta + \sin \theta = m \quad \text{and} \quad \tan \theta - \sin \theta = n \] Now, \( m^2 - n^2 \) is a difference of squares, so we can use the identity \( m^2 - n^2 = (m + n)(m - n) \): \[ m^2 - n^2 = (\tan \theta + \sin \theta + \tan \theta - \sin \theta)(\tan \theta + \sin \theta - \tan \theta + \sin \theta) \] Simplifying: \[ m^2 - n^2 = (2 \tan \theta) \cdot (2 \sin \theta) \] \[ m^2 - n^2 = 4 \cdot \tan \theta \cdot \sin \theta \] Thus, the correct answer is \( 4mn \).