Explanation

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