Find the value of $x$ for which
$sin h^{- 1} x + cos h^{- 1} (x + 2) = 0$

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Solution

${sinh}^{- 1} x + {cosh}^{- 1} (x + 2) = 0$
$∵ {cosh}^{- 1} x = log | x \pm \sqrt{x^{2} + 1} |$
${sinh}^{- 1} x = log | x \pm \sqrt{x^{2} + 1} |$
So, ${cosh}^{- 1} (x + 2) = log | x + 2 \pm \sqrt{{(x + 2)}^{2} + 1} |$
Now, ${sinh}^{- 1} x + {cosh}^{- 1} (x + 2)$
$\Rightarrow log (x + \sqrt{x^{2} + 1}) + log (x + 2 + \sqrt{{(x + 2)}^{2} + 1})$
$\Rightarrow log (x + \sqrt{x^{2} + 1}) + log (x + 2 + \sqrt{x^{2} + 4 + 4 x + 1})$
$\Rightarrow log (x + \sqrt{x^{2} + 1}) (x + 2 + \sqrt{x^{2} + 4 + 4 x + 1})$
$\Rightarrow log (x + \sqrt{x^{2} + 1}) (x + 2 + \sqrt{x^{2} + 4 x .5}) = 0$
$\Rightarrow [(x + 2) + \sqrt{x^{2} + 4 x + 5}] (x + \sqrt{x^{2} + 1})$

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