<s>[INST] Show that two expressions are equivalent I am trying to prove a hyperbolic trigonometric identity and I ran into the following expression:
$$\frac{\left (\sqrt{x^2+1}+x  \right )^2+1}{2\left ( \sqrt{x^2+1} + x \right )} \quad.$$
This expression is supposed to be equivalent to
$$\sqrt{x^2+1} \quad.$$
I tried to algebraically manipulate the original expression to get the required expression and got to:
$$\frac{\left (\sqrt{x^2+1}+x  \right )^2+1}{2\left ( \sqrt{x^2+1} + x \right )} = x+\frac{1}{\sqrt{x^2+1} + x} \quad.$$
but I don't know how I can show that
$$x+\frac{1}{\sqrt{x^2+1} + x} = \sqrt{x^2+1}\quad .$$
 [/INST] What you got is correct . Then, multiply $$\frac{1}{\sqrt{x^2+1}+x}$$ by $$\frac{\sqrt{x^2+1}-x}{\sqrt{x^2+1}-x}(=1).$$
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