Question

Demonstrați indentitatea:​

Answer 1

$\sqrt{t^2+2+2\sqrt{t^2+1}}-\sqrt{t^2+2-2\sqrt{t^2+1}} =\\ \\ = \sqrt{(1+\sqrt{t^2+1})^2}-\sqrt{(1-\sqrt{t^2+1})^2} = \\ \\ = |1+\sqrt{t^2+1}| - |1-\sqrt{t^2+1}| \overset{(*)}{=} \\ \\\\ \sqrt{t^2+1}\geq 1 \Rightarrow \sqrt{t^2+1}-1 \geq 0 \Rightarrow 1-\sqrt{t^2+1}\leq 0 \\\\\Rightarrow |1-\sqrt{t^2+1}| = -(1-\sqrt{t^2+1})\\ \\\\ \overset{(*)}{=} 1+\sqrt{t^2+1} +1-\sqrt{t^2+1} = \\ \\ = \boxed{2}\quad q.e.d.$