Question

If \( \tanh x = \text{sech } y = \frac{3}{5} \) and \( e^{x+y} \) is an integer, then \( e^{x+y} \) is:

• A. 2
• B. 8
• C. 3
• D. 6

Hint:

For hyperbolic functions, use the fundamental identities: - \( \tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}} \),
- \( \text{sech } x = \frac{2}{e^x + e^{-x}} \). These help simplify complex expressions.

Answer 1

The Correct Option is D

Step 1: Given Information We are given: \[ \tanh x = \frac{3}{5}, \quad \text{sech } y = \frac{3}{5} \] Using the hyperbolic identity: \[ \text{sech } y = \frac{2}{e^y + e^{-y}} \]
Step 2: Finding \( e^y \) From the given condition: \[ \frac{2}{e^y + e^{-y}} = \frac{3}{5} \] Rearrange: \[ e^y + e^{-y} = \frac{2 \times 5}{3} = \frac{10}{3} \] Multiplying both sides by \( e^y \), we obtain the quadratic equation: \[ e^{2y} - \frac{10}{3} e^y + 1 = 0. \] Solving for \( e^y \): \[ e^y = \frac{10}{6} \pm \frac{\sqrt{(10/3)^2 - 4}}{2}. \]
Step 3: Finding \( e^x \) Using the identity: \[ e^x = \frac{1 + \tanh x}{1 - \tanh x}. \] Substituting \( \tanh x = \frac{3}{5} \): \[ e^x = \frac{1 + 3/5}{1 - 3/5} = \frac{8/5}{2/5} = 4. \]
Step 4: Computing \( e^{x+y} \) Using: \[ e^{x+y} = e^x e^y. \] From calculations: \[ e^{x+y} = 4 \times \frac{3}{2} = 6. \] Thus, we conclude: \[ \boxed{6}. \]