Question

Rút gọn : \(B=\frac{\sqrt{1-\sqrt{1-x^2}}.\left(\sqrt{\left(1+x\right)^3}+\sqrt{\left(1-x\right)^3}\right)}{2-\sqrt{1-x^2}}\)

Answer 1

ĐKXĐ: \(-1\le x\le1\)

Đặt \(a=\sqrt{1-x}>0\)

\(b=\sqrt{1+x}>0\)

\(\Rightarrow a^2+b^2=2\) và \(a^2-b^2=-2x\)

Khi đó: \(B=\frac{\sqrt{1-ab}\left(a^3+b^3\right)}{2-ab}=\frac{\sqrt{1-ab}\left(a+b\right)\left(a^2+b^2-ab\right)}{2-ab}\)

\(=\frac{\sqrt{1-ab}\left(a+b\right)\left(2-ab\right)}{2-ab}\)\(\Rightarrow B=\sqrt{1-ab}\left(a+b\right)\Rightarrow B\sqrt{2}=\sqrt{2-2ab}\left(a+b\right)\)\(=\sqrt{a^2+b^2-2ab}\left(a+b\right)=\left(a-b\right)\left(a+b\right)=a^2-b^2=\)\(-2x\)

\(\Rightarrow b=-\sqrt{2}x\)