\(A=\left(\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{\left(x-1\right)\left(x+1\right)}}\right).\left(\frac{\sqrt{\left(x-1\right)\left(x+1\right)}}{\sqrt{x+1}-\sqrt{x-1}}\right)=\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}-\sqrt{x-1}}\)
\(=\frac{\left(\sqrt{x+1}+\sqrt{x-1}\right)^2}{2}=\frac{2\left(x+\sqrt{x^2-1}\right)}{2}=x+\sqrt{x^2-1}\)
Thế vào rồi tính nhé
\(\)
Ta có: \(A=\left(\frac{1}{\sqrt{x+1}}+\frac{1}{\sqrt{x-1}}\right):\left(\frac{1}{\sqrt{x+1}}-\frac{1}{\sqrt{x-1}}\right)\) \(\left(ĐK:x\ge1\right)\)
\(\Leftrightarrow A=\left(\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}.\sqrt{x-1}}\right).\left(\frac{\sqrt{x+1}.\sqrt{x-1}}{\sqrt{x+1}-\sqrt{x-1}}\right)\)
\(\Leftrightarrow A=\frac{\left(\sqrt{x+1}+\sqrt{x-1}\right).\left(\sqrt{x+1}-\sqrt{x-1}\right)}{\left(\sqrt{x+1}-\sqrt{x-1}\right)^2}\)
\(\Leftrightarrow A=\frac{x+1-x+1}{x+1+x-1+2\sqrt{\left(x+1\right)\left(x-1\right)}}\)
\(\Leftrightarrow A=\frac{2}{2x+2\sqrt{x^2-1}}\)
Thay \(x=\frac{a^2+b^2}{2ab}\)vào phương trình \(A,\)ta có:
\(A=\frac{1}{\frac{a^2+b^2}{2ab}+\sqrt{\left(\frac{a^2+b^2}{2ab}+1\right)\left(\frac{a^2+b^2}{2ab}-1\right)}}\)
\(\Leftrightarrow A=\frac{1}{\frac{a^2+b^2}{2ab}+\sqrt{\left(\frac{a^2+2ab+b^2}{2ab}\right)\left(\frac{a^2-2ab+b^2}{2ab}\right)}}\)
\(\Leftrightarrow A=\frac{1}{\frac{a^2+b^2}{2ab}+\sqrt{\frac{\left(a+b\right)^2\left(a-b\right)^2}{\left(2ab\right)^2}}}\)
\(\Leftrightarrow A=\frac{1}{\frac{a^2+b^2}{2ab}+\frac{\left(a+b\right)\left(a-b\right)}{2ab}}\)
\(\Leftrightarrow A=\frac{1}{\frac{a^2+b^2+a^2-b^2}{2ab}}\)
\(\Leftrightarrow A=\frac{2ab}{2a^2}\)
\(\Leftrightarrow A=\frac{b}{a}\)
Chúc bn hok tốt
Câu của tạ đức hoàng anh phải là dấu trừ 2 căn (x-1)(x+1) nha
