Упражнение 165

\({\largeа)}\ \left(n+\frac{1}{n}\right)^2=\left(\frac{n^2+1}{n}\right)^2=\frac{ (n^2+1)^2}{n^2}=\frac{n^4+2n^2+1}{n^2};\)

\({\largeб)}\ \left(\frac{a}{b}-\frac{b}{a}\right)^2=\left(\frac{a^2-b^2}{ab}\right)^2=\frac{ (a^2-b^2)^2}{ (ab)^2}=\frac{a^4-2a^2b^2+b^4}{a^2b^2};\)

\({\largeв)}\ \left(\frac{x}{y}+1\right)^2+\left(\frac{x}{y}-1\right)^2=\left(\frac{x+y}{y}\right)^2+\left(\frac{x-y}{y}\right)^2=\frac{ (x+y)^2}{y^2}+\frac{ (x-y)^2}{y^2}=\frac{x^2+2xy+y^2+x^2-2xy+y^2}{y^2}=\frac{2x^2+2y^2}{y^2};\)

\({\largeг)}\ \left(\frac{p}{q}+\frac{q}{p}\right)^2-\left(\frac{p}{q}-\frac{q}{p}\right)^2=\left(\frac{p^2+q^2}{pq}\right)^2-\left(\frac{p^2-q^2}{pq}\right)^2=\frac{ (p^2+q^2)^2}{ (pq)^2}-\frac{ (p^2-q^2)^2}{ (pq)^2}=\frac{p^4+2p^2q^2+q^4-(p^4-2p^2q^2+q^4)}{p^2q^2}=\frac{p^4+2p^2q^2+q^4-p^4+2p^2q^2-q^4}{p^2q^2}=\frac{4p^2q^2}{p^2q^2}=4.\)