\(1+tan^2a=1+\frac{sin^2a}{cos^2a}=\frac{sin^2a+cos^2a}{cos^2a}=\frac{1}{cos^2a}\)
\(1+cot^2a\), không phải \(cos^2a\)
\(1+cot^2a=1+\frac{cos^2a}{sin^2a}=\frac{sin^2a+cos^2a}{sin^2a}=\frac{1}{sin^2a}\)
Cho tam giác ABC có 3 góc nhọn với đường cao AD,BE,CF cắt nhau tại H
a, Cmr : \(\Delta AEF\sim\Delta ABC;\frac{S_{AEF}}{S_{ABC}}=\cos^2A\)
b, Cmr : \(S_{DEF}=\left(1-\cos^2A-\cos^2B-\cos^2C\right).S_{ABC}\)
c, Cmr :\(\frac{HA}{BC}+\frac{HB}{AC}+\frac{HC}{AB}\ge3\)
Cho tam giác ABC có 3 góc nhọn với các đường cao AD,BE,CF cắt nhau tại H.
a, CMR: \(\Delta AEF\sim\Delta ABC\) ; \(\frac{S_{AEF}}{S_{ABC}}=\cos^2\alpha\)
b, CMR: \(S_{DEF}=\left(1-\cos^2A-\cos^2B-\cos^2C\right).S_{ABC}\)
c, Cho biết AH = k.HD. CMR: \(\tan B.\tan C=k+1\)
d, CMR: \(\frac{HA}{BC}+\frac{HB}{AC}+\frac{HC}{AB}\ge\sqrt{3}\)
cho 0<a,b,c<\(\frac{1}{2}\)thỏa mãn a+b+c=1
CMR: \(\frac{1}{a\left(2b+2c-1\right)}+\frac{1}{b\left(2c+2a-1\right)}+\frac{1}{c\left(2a+2b-1\right)}\ge27\)
1. \(\left\{{}\begin{matrix}x,y,z>0\\xyz=1\end{matrix}\right.\) Cmr: \(\frac{x^2}{\left(x+1\right)^2}+\frac{y^2}{\left(y+1\right)^2}+\frac{z^2}{\left(z+1\right)^2}\ge\frac{3}{4}\)\
2. \(a,b,c>0.\) cmr: \(\Sigma\frac{a^3}{\left(2a^2+b^2\right)\left(2a^2+c^2\right)}\le\frac{1}{a+b+c}\)
1.\(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca=3\end{matrix}\right.\) Cmr: \(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge\frac{3}{2}\)
2.\(a,b,c>0\). Cmr: \(\frac{ab^2}{a^2+2b^2+c^2}+\frac{bc^2}{b^2+2c^2+a^2}+\frac{ca^2}{c^2+2a^2+b^2}\le\frac{a+b+c}{4}\)
3. \(a,b,c>0\). Cmr: \(\frac{ab}{a+3b+2c}+\frac{bc}{b+3c+2a}+\frac{ca}{c+3a+2b}\le\frac{a+b+c}{6}\)
Cho a, b, c dương thỏa a +b + c = 3. Cmr: \(\frac{1}{2+a^2b}+\frac{1}{2+b^2c}+\frac{1}{2+c^2a}\ge1\)
Rút gọn biểu thức:
a)\(\frac{2}{\sqrt{5}-\sqrt{3}}+\frac{3}{\sqrt{6}+\sqrt{3}}\)
b)\(\frac{3+2\sqrt{3}}{\sqrt{3}}+\frac{2+\sqrt{2}}{\sqrt{2}+1}-\left(2+\sqrt{3}\right)\)
c)\(\sqrt{3+\sqrt{5}}-\sqrt{3-\sqrt{5}}-\sqrt{2}\)
d)\(\left(1+tan^2a\right)\left(1-sin^2a\right)+\left(1+cotan^2a\right)\left(1-cos^2a\right)\)
CMR: Với mọi góc nhọn \(\alpha\) ta có :
\(a,\sin^2\alpha+\cos^2\alpha=1\)
\(b,\tan\alpha=\frac{\sin\alpha}{\cos\alpha}\)
\(c,\tan^2\alpha+1=\frac{1}{\cos^2\alpha}\)
CMR: \(\frac{2a^3+1}{4b\left(a-b\right)}\ge3\) \(\forall\left\{{}\begin{matrix}a\ge\frac{1}{2}\\\frac{a}{b}>1\end{matrix}\right.\)
