Question

Given that: \[ \frac{1}{\tan A - \tan B} = \]

• A. \( \frac{\sin A \sin B}{\cos(A - B)} \)
• B. \( \frac{\sin A \sin B}{\sin(A - B)} \)
• C. \( \frac{\cos A - \cos B}{\sin A - \sin B} \)
• D. \( \cot A - \cot B \)
• E. \( \frac{\cos A \cos B}{\sin(A - B)} \)

Hint:

When solving problems involving trigonometric identities, use standard identities like \( \tan(A - B) \) and \( \sin(A - B) \), and be mindful of how to manipulate them for simplification.

Answer 1

Answer: (E)

We are given the expression:

\[ \frac{1}{\tan A - \tan B} \]

Using the identity for the tangent of the difference of two angles:

\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]

Rearranging this identity:

\[ \tan A - \tan B = \frac{\sin A \sin B}{\cos(A - B)} \]

So, we can conclude that:

\[ \frac{1}{\tan A - \tan B} = \frac{\cos A \cos B}{\sin(A - B)} \]

Thus, the correct answer is option (E), \( \frac{\cos A \cos B}{\sin(A - B)} \).