<s>[INST] Show that: $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 +1 } )$ could someone Please give me some hint of how to do this question thanks

 [/INST] Hint:
Assuming we already know these hyperbolic functions are invertible:
$$\sinh(\log(x+\sqrt{x^2+1})):=\frac12\left(e^{\log(x+\sqrt{x^2+1})}-e^{-\log(x+\sqrt{x^2+1})}\right)=$$
$$=\frac12\left(x+\sqrt{x^2+1}-\frac1{x+\sqrt{x^2+1}}\right)=\frac12\left(\frac{x^2+x^2+1+2x\sqrt{x^2+1}-1}{x+\sqrt{x^2+1}}\right)=$$
$$=\frac12\left(\frac{2x(x+\sqrt{x^2+1})}{x+\sqrt{x^2+1}}\right)=\frac{2x}2=x$$
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