To simplify the expression [tex]\(\left(x+\frac{1}{x}\right)^2 + \left(x-\frac{1}{x}\right)^2\)[/tex], follow these steps:
1. Expand each squared term separately:
[tex]\(\left(x+\frac{1}{x}\right)^2\)[/tex]:
[tex]\[
\left(x+\frac{1}{x}\right)^2 = x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}
\][/tex]
[tex]\(\left(x-\frac{1}{x}\right)^2\)[/tex]:
[tex]\[
\left(x-\frac{1}{x}\right)^2 = x^2 - 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 = x^2 - 2 + \frac{1}{x^2}
\][/tex]
2. Add the expanded terms together:
[tex]\[
\left(x+\frac{1}{x}\right)^2 + \left(x-\frac{1}{x}\right)^2 = \left(x^2 + 2 + \frac{1}{x^2}\right) + \left(x^2 - 2 + \frac{1}{x^2}\right)
\][/tex]
3. Combine like terms:
[tex]\[
\left(x^2 + 2 + \frac{1}{x^2}\right) + \left(x^2 - 2 + \frac{1}{x^2}\right) = x^2 + x^2 + \frac{1}{x^2} + \frac{1}{x^2} + 2 - 2
\][/tex]
Simplifying further:
[tex]\[
= 2x^2 + \frac{2}{x^2}
\][/tex]
4. Factor out the common term:
[tex]\[
= 2 \left(x^2 + \frac{1}{x^2}\right)
\][/tex]
Thus, the simplified expression for [tex]\(\left(x+\frac{1}{x}\right)^2 + \left(x-\frac{1}{x}\right)^2\)[/tex] is:
[tex]\[
\boxed{\frac{2(x^4 + 1)}{x^2}}
\][/tex]
