2.

\(\Leftrightarrow cos2x-cos8x-sin3x+cos5x-2sin5x.cos5x=0\)

\(\Leftrightarrow2sin5x.sin3x-sin3x+cos5x-2sin5x.cos5x=0\)

\(\Leftrightarrow sin3x\left(2sin5x-1\right)-cos5x\left(2sin5x-1\right)=0\)

\(\Leftrightarrow\left(sin3x-cos5x\right)\left(2sin5x-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cos5x=sin3x=cos\left(\dfrac{\pi}{2}-3x\right)\\sin5x=\dfrac{1}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}5x=\dfrac{\pi}{2}-3x+k2\pi\\5x=3x-\dfrac{\pi}{2}+k2\pi\\5x=\dfrac{\pi}{6}+k2\pi\\5x=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{16}+\dfrac{k\pi}{4}\\x=-\dfrac{\pi}{4}+k\pi\\x=\dfrac{\pi}{30}+\dfrac{k2\pi}{5}\\x=\dfrac{\pi}{6}+\dfrac{k2\pi}{5}\end{matrix}\right.\)

3.

\(\Leftrightarrow1+sinx=cosx-cos3x+2sinx.cosx+1-2sin^2x\)

\(\Leftrightarrow sinx=2sin2x.sinx+2sinx.cosx-2sin^2x\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\Rightarrow x=k\pi\\1=2sin2x+2cosx-2sinx\left(1\right)\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow4sinx.cosx+2cosx-2sinx-1=0\)

\(\Leftrightarrow2cosx\left(2sinx+1\right)-\left(2sinx+1\right)=0\)

\(\Leftrightarrow\left(2cosx+1\right)\left(2sinx-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=\dfrac{1}{2}\\cosx=-\dfrac{1}{2}\end{matrix}\right.\)

\(\Leftrightarrow...\)

4.

\(\Leftrightarrow2sin2x.cosx+sin2x=2cos2x.cosx+cos2x\)

\(\Leftrightarrow sin2x\left(2cosx+1\right)=cos2x\left(2cosx+1\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}2cosx+1=0\\sin2x=cos2x\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}cosx=-\dfrac{1}{2}\\tan2x=1\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\pm\dfrac{2\pi}{3}+k2\pi\\x=\dfrac{\pi}{8}+\dfrac{k\pi}{2}\\\end{matrix}\right.\)

is not different

is not different 

\(y'=\dfrac{\left(-2x+2\right)\left(x-3\right)-\left(-x^2+2x+c\right)}{\left(x-3\right)^2}=\dfrac{-x^2+6x-6-c}{\left(x-3\right)^2}\)

\(\Rightarrow\) Cực đại và cực tiểu của hàm là nghiệm của: \(-x^2+6x-6-c=0\) (1)

\(\Delta'=9-\left(6+c\right)>0\Rightarrow c< 3\)

Gọi \(x_1;x_2\) là 2 nghiệm của (1) \(\Rightarrow\left\{{}\begin{matrix}-x_1^2+6x_1-6=c\\-x_2^2+6x_2-6=c\end{matrix}\right.\)

\(\Rightarrow m-M=\dfrac{-x_1^2+2x_1+c}{x_1-3}-\dfrac{-x_2^2+2x_2+c}{x_2-3}=4\)

\(\Leftrightarrow\dfrac{-2x_1^2+8x_1-6}{x_1-3}-\dfrac{-2x_2^2+8x_2-6}{x_2-3}=4\)

\(\Leftrightarrow2\left(1-x_1\right)-2\left(1-x_2\right)=4\)

\(\Leftrightarrow x_2-x_1=2\)

Kết hợp với Viet: \(\left\{{}\begin{matrix}x_2-x_1=2\\x_1+x_2=6\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x_1=2\\x_2=4\end{matrix}\right.\)

\(\Rightarrow c=2\)

Có 1 giá trị nguyên

thay \(x=3-2\sqrt{2}\) vào P ta có:

\(\dfrac{x+8}{\sqrt{x}+1}=\dfrac{3-2\sqrt{2}+8}{\sqrt{3-2\sqrt{2}}+1}=\dfrac{11-2\sqrt{2}}{\sqrt{2}-1+1}=\dfrac{11-2\sqrt{2}}{\sqrt{2}}\)

\(b,x=3-2\sqrt{2}=\left(\sqrt{2}-1\right)^2\)

Thay vào P, ta được:

\(P=\dfrac{3-2\sqrt{2}+8}{\sqrt{\left(\sqrt{2}-1\right)^2}+1}=\dfrac{11-2\sqrt{2}}{\sqrt{2}}=\dfrac{11\sqrt{2}-4}{2}\)

b: Thay \(x=3+2\sqrt{2}\) vào P, ta được:

\(P=\dfrac{3+2\sqrt{2}+8}{2+\sqrt{2}}=\dfrac{18-7\sqrt{2}}{2}\)

Bạn nên chịu khó gõ đề ra khả năng được giúp sẽ cao hơn.

Câu h của em đây nhé

h, ( 1 + \(\dfrac{3-\sqrt{3}}{\sqrt{3}-1}\)).(1 - \(\dfrac{3+\sqrt{3}}{\sqrt{3}+1}\))

= \(\dfrac{\sqrt{3}-1+3-\sqrt{3}}{\sqrt{3}-1}\).\(\dfrac{\sqrt{3}+1-3-\sqrt{3}}{\sqrt{3}+1}\)

= \(\dfrac{2}{\sqrt{3}-1}\).\(\dfrac{-2}{\sqrt{3}+1}\)

= \(\dfrac{-4}{2}\)

= -2