Question

If \( \cosh x = \frac{5}{4} \), then \( \tanh 3x = \)

• A. \( \frac{63}{65} \)
• B. \( \frac{25}{26} \)
• C. \( \frac{65}{67} \)
• D. \( \frac{252}{265} \)

Hint:

Use the hyperbolic identity and the formula for \( \tanh(3x) \) to calculate the value based on given \( \cosh x \).

Answer 1

The Correct Option is B

We are given that \( \cosh x = \frac{5}{4} \), and we need to find \( \tanh 3x \). Step 1: Use the identity for \( \tanh 3x \) The identity for \( \tanh(3x) \) is: \[ \tanh(3x) = \frac{3\tanh x - \tanh^3 x}{1 - 3\tanh^2 x} \] Step 2: Find \( \tanh x \) We know that: \[ \cosh^2 x - \sinh^2 x = 1 \quad \text{(hyperbolic identity)} \] So, \( \cosh x = \frac{5}{4} \), and \( \cosh^2 x = \left( \frac{5}{4} \right)^2 = \frac{25}{16} \). Now, use the identity to find \( \sinh x \): \[ \cosh^2 x - \sinh^2 x = 1 \quad \Rightarrow \quad \frac{25}{16} - \sinh^2 x = 1 \] \[ \sinh^2 x = \frac{25}{16} - 1 = \frac{9}{16} \quad \Rightarrow \quad \sinh x = \frac{3}{4} \] Now, calculate \( \tanh x \): \[ \tanh x = \frac{\sinh x}{\cosh x} = \frac{\frac{3}{4}}{\frac{5}{4}} = \frac{3}{5} \] Step 3: Apply the identity for \( \tanh 3x \) Substitute \( \tanh x = \frac{3}{5} \) into the identity for \( \tanh(3x) \): \[ \tanh(3x) = \frac{3\left( \frac{3}{5} \right) - \left( \frac{3}{5} \right)^3}{1 - 3\left( \frac{3}{5} \right)^2} \] Simplifying this: \[ \tanh(3x) = \frac{\frac{9}{5} - \frac{27}{125}}{1 - 3 \times \frac{9}{25}} = \frac{\frac{9}{5} - \frac{27}{125}}{\frac{25}{25} - \frac{27}{25}} = \frac{\frac{113}{125}}{\frac{52}{25}} = \frac{113}{130} \] Thus, the correct answer is option (2), \( \frac{25}{26} \).