- Xét \(sin\frac{x}{5}=0\Rightarrow C=...\)

- Với \(sin\frac{x}{5}\ne0\)

\(C.sin\frac{x}{5}=sin\frac{x}{5}.cos\frac{x}{5}.cos\frac{2x}{5}cos\frac{4x}{5}cos\frac{8x}{5}\)

\(=\frac{1}{2}sin\frac{2x}{5}cos\frac{2x}{5}cos\frac{4x}{5}cos\frac{8x}{5}\)

\(=\frac{1}{4}sin\frac{4x}{5}cos\frac{4x}{5}cos\frac{8x}{5}=\frac{1}{8}sin\frac{8x}{5}cos\frac{8x}{5}\)

\(=\frac{1}{16}sin\frac{16x}{5}\Rightarrow C=\frac{sin\frac{16x}{5}}{16.sin\frac{x}{5}}\)

\(D=sin\frac{x}{7}+sin\frac{5x}{7}+2sin\frac{3x}{7}\)

\(=2sin\frac{3x}{7}cos\frac{2x}{7}+2sin\frac{3x}{7}\)

\(=2sin\frac{3x}{7}\left(cos\frac{2x}{7}+1\right)=4cos^2\frac{x}{7}.sin\frac{3x}{7}\)

\(A=cos\frac{\pi}{7}cos\frac{3\pi}{7}cos\frac{5\pi}{7}=cos\frac{\pi}{7}cos\frac{4\pi}{7}cos\frac{2\pi}{7}\)

\(\Rightarrow A.sin\frac{\pi}{7}=sin\frac{\pi}{7}.cos\frac{\pi}{7}.cos\frac{2\pi}{7}cos\frac{4\pi}{7}\)

\(=\frac{1}{2}sin\frac{2\pi}{7}cos\frac{2\pi}{7}cos\frac{4\pi}{7}=\frac{1}{4}sin\frac{4\pi}{7}cos\frac{4\pi}{7}\)

\(=\frac{1}{8}sin\frac{8\pi}{7}=\frac{1}{8}sin\left(\pi+\frac{\pi}{7}\right)=-\frac{1}{8}sin\frac{\pi}{7}\)

\(\Rightarrow A=-\frac{1}{8}\)

\(B=sin6.cos48.cos24.cos12\)

\(B.cos6=sin6.cos6.cos12.cos24.cos48\)

\(=\frac{1}{2}sin12.cos12.cos24.cos48=\frac{1}{4}sin24.cos24.cos48\)

\(=\frac{1}{8}sin48.cos48=\frac{1}{16}sin96\)

\(=\frac{1}{16}sin\left(90+6\right)=\frac{1}{16}cos6\Rightarrow B=\frac{1}{16}\)

Chỉ đúng trong trường hợp các số thực dương (kì lạ là các bạn rất thích quên điều kiện này khi đăng đề lên)

a/ \(\frac{a^3}{b^2}+a\ge2\sqrt{\frac{a^4}{b^2}}=\frac{2a^2}{b}\) ; \(\frac{b^3}{c^2}+b\ge\frac{2b^2}{c}\); \(\frac{c^3}{a^2}+c\ge\frac{2c^2}{a}\)

Cộng vế với vế:

\(VT+a+b+c\ge2VP\Rightarrow VT\ge2VP-\left(a+b+c\right)\)

Mà \(2VP=\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+\frac{\left(a+b+c\right)^2}{a+b+c}\)

\(\Rightarrow2VP\ge VP+a+b+c\)

\(\Rightarrow2VP-\left(a+b+c\right)\ge VP\)

\(\Rightarrow VT\ge VP\)

Dấu "=" xảy ra khi \(a=b=c\)

Câu dưới tương tự:

\(\frac{a^5}{b^3}+a^2+a^2\ge\frac{3a^3}{b}\) , làm tương tự với 2 cái còn lại và cộng lại:

\(\Rightarrow VT+2\left(a^2+b^2+c^2\right)\ge3\left(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\right)=3\left(\frac{a^4}{ab}+\frac{b^4}{ca}+\frac{c^4}{ab}\right)\ge\frac{3\left(a^2+b^2+c^2\right)^2}{ab+bc+ca}\ge3\left(a^2+b^2+c^2\right)\)

\(\Rightarrow VT\ge a^2+b^2+c^2\)

Dấu "=" xảy ra khi \(a=b=c\)

\(\frac{\sin^4\alpha}{a}+\frac{\cos^4\alpha}{b}\ge\frac{\left(\sin^2\alpha+\cos^2\alpha\right)^2}{a+b}=\frac{1}{a+b}\)

\("="\Leftrightarrow\frac{\sin^2\alpha}{a}=\frac{\cos^2\alpha}{b}\Leftrightarrow\sin^2\alpha.b=a-a.\sin^2\alpha\)

\(\Leftrightarrow\sin^2\alpha\left(b+a\right)=a\Rightarrow\sin^2\alpha=\frac{a}{a+b}\)

\(\cos^2\alpha.a=b-b\cos^2\alpha\Rightarrow\cos^2\alpha=\frac{b}{a+b}\)

\(\Rightarrow M=\frac{\frac{a^5}{\left(a+b\right)^5}}{a^4}+\frac{\frac{b^5}{\left(a+b\right)^5}}{b^4}=\frac{a+b}{\left(a+b\right)^5}=\frac{1}{\left(a+b\right)^4}\) => D