Correct Answer - Option 3 :
\(\frac{15}{17}\)
Formula used:
1. cosh x = \(\frac{e^x\ +\ e^{-x}}{2}\)
2. sinh x = \(\frac{e^x\ -\ e^{-x}}{2}\)
3. sinh x + cosh x = ex
4. cosh2x - sinh2x = 1
5. \(tanh\ 2x\ =\ \frac{2tanh\ x}{1\ +\ tanh^2\ x}\)
6. \(tanh\ x\ = \ \frac{sinh\ x}{cosh\ x}\)
Calculation:
Given that,
\(sinh\ x\ =\ \frac{3}{4}\) ----(1)
we know that,
cosh2x - sinh2x = 1
⇒ cosh2x = sinh2x + 1
⇒ cosh2x = \((\frac{3}{4})^2\) + 1
⇒ cosh x = 5/4 -----(2)
Divide equation (1) and (2)
tanh x = 3/5
We know that, \(tanh\ 2x\ =\ \frac{2tanh\ x}{1\ +\ tanh^2\ x}\)
⇒ \(tanh\ 2x\ =\ \frac{2(\frac{3}{5})}{1\ +\ (\frac{3}{5})^2}\)
⇒ \(tanh\ 2x\ =\ \frac{15}{17}\)
Hence, option 3 is correct.
