Question

If \( \alpha = \log_e(2 + \sqrt{3}) \), then evaluate: \[ \frac{\cosh \alpha }{1 - \tanh \alpha} + \frac{\sinh \alpha}{1 - \coth \alpha} \]

• A. \( 4 + 2\sqrt{3} \)
• B. \( 7 + 4\sqrt{3} \)
• C. \( \frac{\sqrt{3} + 1}{2} \)
• D. \( 2 + \sqrt{3} \)

Hint:

When working with hyperbolic functions, remember the basic identities for \( \cosh \), \( \sinh \), \( \tanh \), and \( \coth \).

Answer 1

The Correct Option is D

We are given that \( \alpha = \log_e(2 + \sqrt{3}) \). Using the following standard identities: \[ \cosh \alpha = \frac{e^\alpha + e^{-\alpha}}{2}, \quad \sinh \alpha = \frac{e^\alpha - e^{-\alpha}}{2} \] \[ \tanh \alpha = \frac{\sinh \alpha}{\cosh \alpha}, \quad \coth \alpha = \frac{\cosh \alpha}{\sinh \alpha} \] We can substitute these into the given expression: \[ \frac{\cosh \alpha + \sinh \alpha}{1 - \tanh \alpha} + \frac{1}{1 - \coth \alpha} = 2 + \sqrt{3} \] Thus, the correct answer is \( 2 + \sqrt{3} \), which corresponds to option (4).