do AD//CM nên \(\frac{AD}{CM}=\frac{BA}{BM}=\frac{c}{b+c}\)

mà \(CM< AM+AC=2b=>\frac{c}{bc}>\frac{AD}{2b}=>\frac{1}{l_a}>\frac{1}{2}\left(\frac{1}{b}+\frac{1}{c}\right)\left(1\right)\)

tương tự ta có 

\(\hept{\begin{cases}\frac{1}{l_b}>\frac{1}{2}\left(\frac{1}{c}+\frac{1}{a}\right)\left(2\right)\\\frac{1}{l_c}>\frac{1}{2}\left(\frac{1}{b}+\frac{1}{c}\right)\left(3\right)\end{cases}}\)

cộng (1) (2) (3) zế zới zế ta được đpcm

Sai chỗ nào tự sửa nha :)))

Bài này hình như trong sách nào mà t quên ròi, ai nhớ nhắc với

file:///C:/Users/THAOCAT/Pictures/%C4%90%E1%BA%A1i%20S%E1%BB%91%20-%20H%C3%ACnh%20H%E1%BB%8Dc%20L%E1%BB%9Bp%208,9/%C4%90%E1%BB%81%20thi%20hsg%20to%C3%A1n%208/De%20thi%20chon%20HSG.pdf

Tự chứng minh từng cái này rồi suy ra cái đó nhé b.

Ta có: \(sin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}-sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}=sin^2\frac{A}{2}\)

Tương tự ta suy ra: 

\(sin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}+cos\frac{A}{2}sin\frac{B}{2}cos\frac{C}{2}+cos\frac{A}{2}cos\frac{B}{2}sin\frac{C}{2}=sin^2\frac{A}{2}+sin^2\frac{B}{2}+sin^2\frac{C}{2}+3sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}\left(1\right)\)

Tiếp theo chứng minh:

\(2sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}=\frac{cosA+cosB+cosC-1}{2}\left(2\right)\)

\(sin^2\frac{A}{2}+sin^2\frac{B}{2}+sin^2\frac{C}{2}=\frac{3}{2}-\frac{cosA+cosB+cosC}{2}\left(3\right)\)

\(tan\frac{A}{2}tan\frac{B}{2}+tan\frac{B}{2}tan\frac{C}{2}+tan\frac{C}{2}tan\frac{A}{2}=1\left(4\right)\)

Từ (1), (2), (3), (4) suy được điều phải chứng minh

ko hiểu ( vì em mới học lớp 6)

trinh le na

cho bạn 4 năm nữa cũng chưa hiểu đâu

\(A+B+C=180^0\Rightarrow A+B=180^0-C\)

\(\Rightarrow sin\left(A+B\right)=sin\left(180^0-C\right)=sinC\)

\(cos\left(A+B\right)=cos\left(180^0-C\right)=-cosC\)

\(tan\left(A+B\right)=tan\left(180^0-C\right)=-tanC\)

b/ \(\frac{A+B+C}{2}=90^0\Rightarrow\frac{A+B}{2}=90^0-\frac{C}{2}\)

\(\Rightarrow sin\frac{A+B}{2}=sin\left(90^0-\frac{C}{2}\right)=cos\frac{C}{2}\)

\(cos\frac{A+B}{2}=cos\left(90^0-\frac{C}{2}\right)=sin\frac{C}{2}\)

\(tan\frac{A+B}{2}=tan\left(90-\frac{C}{2}\right)=cot\frac{C}{2}\)

c/ \(A+B=180^0-C\Rightarrow tan\left(A+B\right)=-tanC\)

\(\Leftrightarrow\frac{tanA+tanB}{1-tanA.tanB}=-tanC\)

\(\Leftrightarrow tanA+tanB=-tanC+tanA.tanB.tanC\)

\(\Leftrightarrow tanA+tanB+tanC=tanA.tanB.tanC\)

d/ \(sinA+sinB+sinC=2sin\frac{A+B}{2}cos\frac{A-B}{2}+2sin\frac{C}{2}.cos\frac{C}{2}\)

\(=2cos\frac{C}{2}.cos\frac{A-B}{2}+2sin\frac{C}{2}.cos\frac{C}{2}\)

\(=2cos\frac{C}{2}\left(cos\frac{A-B}{2}+sin\frac{C}{2}\right)\)

\(=2cos\frac{C}{2}\left(cos\frac{A-B}{2}+cos\frac{A+B}{2}\right)\)

\(=4cos\frac{C}{2}.cos\frac{A}{2}.cos\frac{B}{2}\)

e/

\(cosA+cosB+cosC=2cos\frac{A+B}{2}cos\frac{A-B}{2}+1-2sin^2\frac{C}{2}\)

\(=1+2sin\frac{C}{2}.cos\frac{A-B}{2}-2sin^2\frac{C}{2}\)

\(=1+2sin\frac{C}{2}\left(cos\frac{A-B}{2}-sin\frac{C}{2}\right)\)

\(=1+2sin\frac{C}{2}\left(cos\frac{A-B}{2}-cos\frac{A+B}{2}\right)\)

\(=1+4sin\frac{C}{2}.sin\frac{A}{2}sin\frac{B}{2}\)

f/

\(sin2A+sin2B+sin2C=2sin\left(A+B\right).cos\left(A-B\right)+2sinC.cosC\)

\(=2sinC.cos\left(A-B\right)+2sinC.cosC\)

\(=2sinC\left(cos\left(A-B\right)+cosC\right)\)

\(=2sinC\left[cos\left(A-B\right)-cos\left(A+B\right)\right]\)

\(=4sinC.sinA.sinB\)

g/

\(cos^2A+cos^2B+cos^2C=\frac{1}{2}+\frac{1}{2}cos2A+\frac{1}{2}+\frac{1}{2}cos2B+cos^2C\)

\(=1+\frac{1}{2}\left(cos2A+cos2B\right)+cos^2C\)

\(=1+cos\left(A+B\right).cos\left(A-B\right)+cos^2C\)

\(=1-cosC.cos\left(A-B\right)+cos^2C\)

\(=1-cosC\left(cos\left(A-B\right)-cosC\right)\)

\(=1-cosC\left[cos\left(A-B\right)+cos\left(A+B\right)\right]\)

\(=1-2cosC.cosA.cosB\)