Podmínky:
I.\ a\ – \frac{1}{a} \ne 0 \to a \ne \frac{1}{a} \to a^{2} \ne 1 \to |a| \ne \sqrt 1 \to a \ne \pm 1;0
II.\ a + \frac{1}{a} \ne 0 \to a \ne\ –\ \frac{1}{a} \to a^{2} \ne\ – 1 \to a \in \mathbb{R} – \{0\}
III.\ \left (a + \frac{1}{a} \right )^{2} – \left (a\ – \frac{1}{a} \right )^{2}\neq 0\to \left (a + \frac{1}{a} \right )^{2}\neq \left (a\ – \frac{1}{a} \right )^{2}\to
\to (\frac{a^2 + 1}{a})^2 \ne\ (\frac{a^2 – 1}{a})^2 \to (a^2 + 1)^2 \ne\ (a^2 –1)^2 \to |a^2 + 1|\ne\ |a^2 –1|
a)\ a^2 + 1 \ne a^2 –\ 1 \to 1\ne\ –\ 1 \to a \in \mathbb{R}\ – \{0\}
b)\ a^2 + 1 \ne\ –(a^2 –\ 1) \to 2 a^2 \ne 0 \to a \in \mathbb{R}\ – \{0\}