Ta có \(AB=AD\Rightarrow\Delta ABD\)vuông  cân tại A 

\(\Rightarrow\widehat{ADI}=45^0\Rightarrow\widehat{EID}=45^0\Rightarrow\Delta IED\)vuông cân tại \(E\Rightarrow IE=ED\)

Xét \(\Delta ABD\)có \(IE\)song song \(AB\Rightarrow\frac{IB}{ID}=\frac{AE}{ED}\)

Mà \(IE=ED\Rightarrow\frac{IB}{ID}=\frac{AE}{IE}\left(đpcm\right)\)

b. Ta có \(AB^2=BH.BC;AC^2=CH.BC\Rightarrow\frac{AB^2}{AC^2}=\frac{BH}{HC}\)

Có \(\widehat{B}+\widehat{C}=90^0;\widehat{BAH}+\widehat{B}=90^0\Rightarrow\widehat{BAH}=\widehat{C}\)

Lại có \(\widehat{BAH}=\widehat{AIE}\)Vì 2 góc ở vị trí so le trong \(\Rightarrow\widehat{C}=\widehat{AIE}\)

Xét \(\Delta ABC\)và \(\Delta EAI\)

có \(\hept{\begin{cases}\widehat{A}=\widehat{E}=90^0\\\widehat{C}=\widehat{AIE}\left(cmt\right)\end{cases}\Rightarrow\Delta ABC~\Delta EAI\left(g-g\right)}\)

\(\Rightarrow\frac{AB}{AC}=\frac{AE}{IE}\)

Lại có \(\frac{AE}{EI}=\frac{IB}{ID}\Rightarrow\frac{IB}{ID}=\frac{AB}{AC}\Rightarrow\frac{IB^2}{ID^2}=\frac{HB}{HC}\left(đpcm\right)\)

ta có: tan x.cot x=1=>cot x=2/5

1+tan^2 x=1/cos^2 x=>cos=2/\(\sqrt{29}\)

tan x=sin x/cos x=>sin x=\(\frac{5\sqrt{29}}{4}\)

\(=\frac{x+2}{\left(\sqrt{x}\right)^3-1}+\frac{\sqrt{x}+1}{x+\sqrt{x}+1}+\frac{1}{1-\sqrt{x}}\)

\(=\frac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\frac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(=\frac{x+2+x-1-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(=\frac{\sqrt{x}}{x+\sqrt{x}+1}\)

Có :

\(\frac{a^2+5}{\sqrt{a^2+4}}\ge2\)     (sửa lại đề)

\(\Rightarrow a^2+5\ge2.\sqrt{a^2+4}\)

\(\Rightarrow\left(a^2+5\right)^2\ge4.\left(a^2+4\right)\)

\(\Rightarrow a^4+10a^2+25\ge4.a^2+16\)

\(\Rightarrow a^4+6a^2+9\ge0\)

\(\Rightarrow\left(a^2+3\right)^2\ge0\)  (Cái này đúng)

=> BĐT cần chứng minh là đúng .

đề sai rồi, giả sử a=0 thì \(\frac{a^2+5}{\sqrt{a^2}+4}=\frac{5}{4}=1,25< 2\) 

Có nghĩa là gì thế bạn??????????????????

?_?

là sao mình k hiểu