Q64 / 160 · AP EAMCET 2023 (15 May Shift 2) · Mathematics

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

  • Correct answer
    A
    $\frac{63}{65}$
  • B
    $\frac{25}{26}$
  • C
    $\frac{65}{67}$
  • D
    $\frac{252}{265}$

Solution

Given, $\cosh x=\frac{5}{4}$ $\Rightarrow e^x+e^{-x}=\frac{5}{2}$ Now, $\left(e^x+e^{-x}\right)^3=e^{3 x}+e^{-3 x}+3 e^x e^{-x}\left(e^x+e^{-x}\right)$ $\Rightarrow \frac{125}{8}=e^{3 x}+e^{-3 x}+\frac{15}{2} \Rightarrow e^{3 x}+e^{-3 x}=\frac{65}{8}$ and $\cosh ^2 x-\sinh ^2 x=1 \Rightarrow \sinh x=\frac{3}{4}$ $\begin{aligned} & \Rightarrow e^x-e^{-x}=\frac{3}{2} \\ & \left(e^x-e^{-x}\right)^3=e^{3 x}-e^{-3 x}-3 e^x \cdot e^{-x}\left(e^x-e^{-x}\right) \\ & \Rightarrow \frac{27}{8}=e^{3 x}-e^{-3 x}-3 \times \frac{5}{2} \Rightarrow e^{3 x}-e^{-3 x}=\frac{63}{8} \end{aligned}$ Now, $\tanh 3 x=\frac{\sinh 3 x}{\cosh 3 x}=\frac{63}{8} \times \frac{8}{65}=\frac{63}{65}$

Hence, the correct answer is option: 'A'.

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