Question

If \( \tan A = \tan \alpha \coth x = \cot \beta \tanh x \), then \( \tan(\alpha + \beta) = \):

• A. \( \cosh 2x \csc 2A \)
• B. \( \cosh 2x \sec 2A \)
• C. \( \sinh 2x \cos 2A \)
• D. \( \sinh 2x \csc 2A \)

Hint:

For expressions involving hyperbolic and trigonometric functions, use identities such as \( \sinh 2x = 2 \sinh x \cosh x \) and simplify step by step.

Answer 1

The Correct Option is D

We are given that \( \tan A = \tan \alpha \coth x = \cot \beta \tanh x \). Step 1: Use the identity for the addition of tangents: \[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \] Step 2: Substitute the expressions for \( \tan \alpha \) and \( \tan \beta \) using the given relations, and simplify using hyperbolic and trigonometric identities. Step 3: The final result simplifies to \( \sinh 2x \csc 2A \). Thus, the correct answer is \( \sinh 2x \csc 2A \).