Question

If \(\cosec\, x = \frac{4}{5}\), then \( \cosh x = \) ?

• A. \( \sqrt{\frac{41}{21}} \)
• B. \( \sqrt{\frac{41}{19}} \)
• C. \( \sqrt{\frac{41}{25}} \)
• D. \( \sqrt{\frac{41}{16}} \)

Hint:

In hyperbolic identities, use \( \cosh^2 x - \sinh^2 x = 1 \) (similar to \( \cos^2 x + \sin^2 x = 1 \) in trigonometry).

Answer 1

The Correct Option is D

To solve for \( \cosh x \) given \(\cosech\, x = \frac{4}{5}\), we start by recalling the relationship \(\cosech\, x = \frac{1}{\sinh x}\). Therefore, \(\sinh x = \frac{5}{4}\).

We know the hyperbolic identity \(\cosh^2 x - \sinh^2 x = 1\). Plugging in the value of \(\sinh x\), we have:

\(\cosh^2 x - \left(\frac{5}{4}\right)^2 = 1\)

\(\cosh^2 x - \frac{25}{16} = 1\)

\(\cosh^2 x = 1 + \frac{25}{16}\)

To add these, convert 1 to \(\frac{16}{16}\):

\(\cosh^2 x = \frac{16}{16} + \frac{25}{16} = \frac{41}{16}\)

Thus, \(\cosh x = \sqrt{\frac{41}{16}} = \frac{\sqrt{41}}{4}\).

The correct choice is \(\sqrt{\frac{41}{16}}\).