Question

If \( \cosh 2x = 199 \), then \( \cot hx = \)

• A. \( \dfrac{5}{3\sqrt{11}} \)
• B. \( \dfrac{5}{6\sqrt{11}} \)
• C. \( \dfrac{7}{3\sqrt{11}} \)
• D. \( \dfrac{10}{3\sqrt{11}} \)

Hint:

Use the identity \( \cosh(2x) = 2\sinh^2 x + 1 \) and \( \cosh^2 x - \sinh^2 x = 1 \) to relate hyperbolic functions.

Answer 1

The Correct Option is D

Given: \[ \cosh(2x) = 199 \] We know the identity: \[ \cosh(2x) = 2\sinh^2 x + 1 \Rightarrow 2\sinh^2 x = 198 \Rightarrow \sinh^2 x = 99 \] Also, \[ \cosh^2 x - \sinh^2 x = 1 \Rightarrow \cosh^2 x = 100 \Rightarrow \cosh x = \sqrt{100} = 10 \] So, \[ \cot hx = \frac{\cosh x}{\sinh x} = \frac{10}{\sqrt{99}} = \frac{10}{3\sqrt{11}} \]