Question

If \(\sinh x = \dfrac{\sqrt{21}}{2}\) then \(\cosh 2x + \sinh 2x = \)

• A. \(\dfrac{21}{2}\)
• B. \(\dfrac{25}{2}\)
• C. \(\dfrac{23 + 5\sqrt{21}}{2}\)
• D. \(\dfrac{32 + 5\sqrt{23}}{2}\)

Hint:

To calculate \(\cosh 2x + \sinh 2x\), use the standard hyperbolic identities and substitute known values for \(\sinh x\) and \(\cosh x\).

Answer 1

The Correct Option is C

Step 1: Use the hyperbolic identities.
We are given the following hyperbolic function identities: - \( \cosh^2(x) = \cosh^2(x) + \sinh^2(x) \) - \( \sinh(2x) = 2 \sinh(x) \cosh(x) \) Also, we are given: \[ \sinh(x) = \frac{\sqrt{21}}{2} \] 

Step 2: Use the identity for \( \cosh^2(x) \).
Using the identity \( \cosh^2(x) - \sinh^2(x) = 1 \), we can solve for \( \cosh^2(x) \): \[ \cosh^2(x) = 1 + \sinh^2(x) \] 
Substituting \( \sinh(x) = \frac{\sqrt{21}}{2} \): \[ \cosh^2(x) = 1 + \left(\frac{\sqrt{21}}{2}\right)^2 = 1 + \frac{21}{4} = \frac{25}{4} \] Thus, \( \cosh(x) = \frac{5}{2} \). 

Step 3: Calculate \( \cosh(2x) \) and \( \sinh(2x) \).
Now, use the identities to calculate \( \cosh(2x) \) and \( \sinh(2x) \): \[ \cosh(2x) = \cosh^2(x) + \sinh^2(x) = \frac{25}{4} + \frac{21}{4} = \frac{46}{4} = \frac{23}{2} \] \[ \sinh(2x) = 2 \sinh(x) \cosh(x) = 2 \times \frac{\sqrt{21}}{2} \times \frac{5}{2} = \frac{5 \sqrt{21}}{2} \]

 Conclusion:
Thus, the final result is: \[ \cosh(2x) + \sinh(2x) = \frac{23}{2} + \frac{5 \sqrt{21}}{2} \]