Q67 / 160 · AP EAMCET 2009 · Mathematics

$\sinh ^{-1} 2+\sinh ^{-1} 3=x \Rightarrow \cosh x$ is equal to

  • A
    $\frac{1}{2}(3 \sqrt{5}+2 \sqrt{10})$
  • B
    $\frac{1}{2}(3 \sqrt{5}-2 \sqrt{10})$
  • Correct answer
    C
    $\frac{1}{2}(12+2 \sqrt{50})$
  • D
    $\frac{1}{2}(12-2 \sqrt{50})$

Solution

$\begin{aligned} & \text { Given, } \sinh ^{-1} 2+\sinh ^{-1} 3=x \\ & \Rightarrow \quad \cosh \left(\sinh ^{-1} 2+\sinh ^{-1} 3\right)=\cosh x \\ & \Rightarrow \quad \cosh \left(\sinh ^{-1} 2\right) \cosh \left(\sinh ^{-1} 3\right) \\ & +\sinh \left(\sinh ^{-1} 2\right) \sinh \left(\sinh ^{-1} 3\right) \\ & =\cosh x \\ & \Rightarrow \cosh x=\cosh \left(\cosh ^{-1} \sqrt{1+2^2}\right) \times \\ & \cosh \left(\cosh ^{-1} \sqrt{1+3^2}\right)+2 \times 3 \\ & \Rightarrow \cosh x=(\sqrt{5} \sqrt{10}+6) \times \frac{2}{2} \\ & =\frac{1}{2}(12+2 \sqrt{50}) \\ & \end{aligned}$

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

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