Q64 / 160 · AP EAMCET 2023 (16 May Shift 1) · Mathematics

$\cosh \alpha+\sinh \alpha=\mathrm{e}^3$ and $\sinh \mathrm{x}=\frac{\alpha}{\alpha+1}$, then $\tanh \mathrm{x}=$

  • Correct answer
    A
    $\frac{\alpha}{\alpha+2}$
  • B
    $\frac{\alpha}{\alpha-3}$
  • C
    $\frac{\alpha}{\alpha+4}$
  • D
    $\frac{2 \alpha}{\alpha-1}$

Solution

Since, $\cos h \alpha+\sin h \alpha=e^3$ $\Rightarrow \frac{e^\alpha+e^{-\alpha}}{2}+\frac{e^\alpha-e^{-\alpha}}{2}=e^3 \Rightarrow \alpha=3$ Now, $\sin h x=\frac{\alpha}{\alpha+1}=\frac{3}{4}$ $\begin{aligned} & \Rightarrow \cos h x=\sqrt{1+\sin h^2 x}=\frac{5}{4} \\ & \Rightarrow \tan h x=\frac{\sin h x}{\cos h x}=\frac{3}{5}=\frac{\alpha}{\alpha+2}\end{aligned}$

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

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