ĐK: \(x\ne\dfrac{\pi}{2}+k\pi\)

Ta có: 

\(\left\{{}\begin{matrix}tanx=3\\sin^2x+cos^2x=1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}sinx=3cosx\\9cos^2x+cos^2x=1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}sinx=3cosx\\cos^2x=\dfrac{1}{10}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}sinx=3cosx\\cosx=\pm\dfrac{1}{\sqrt{10}}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}sinx=\dfrac{3}{\sqrt{10}}\\cosx=\dfrac{1}{\sqrt{10}}\end{matrix}\right.\\\left\{{}\begin{matrix}sinx=-\dfrac{3}{\sqrt{10}}\\cosx=-\dfrac{1}{\sqrt{10}}\end{matrix}\right.\end{matrix}\right.\)

Ta có \(\tan x=\frac{1}{2}\Rightarrow\frac{\sin x}{\cos x}=\frac{1}{2}\Rightarrow\cos x=2\sin x\)

Từ đó \(\frac{\cos x+\sin x}{\cos x-\sin x}=\frac{2\sin x+\sin x}{2\sin x-\sin x}=\frac{3\sin x}{\sin x}=3\)

Vậy \(\frac{\cos x+\sin x}{\cos x-\sin x}=3\)

đáp án không giống lắm 

2: \(\left(sinx+cosx\right)^2=1+2\cdot sinx\cdot cosx=1+2\cdot\dfrac{\sqrt{3}}{4}=1+\dfrac{\sqrt{3}}{2}=\dfrac{2+\sqrt{3}}{2}\)

=>\(sinx+cosx=\dfrac{\sqrt{3}+1}{2}\)

mà sin x*cosx=căn 3/4

nên sinx,cosx là các nghiệm của phương trình là:

\(a^2-\dfrac{\sqrt{3}+1}{2}\cdot a+\dfrac{\sqrt{3}}{4}=0\)

=>\(\left[{}\begin{matrix}a=\dfrac{\sqrt{3}}{2}\\a=\dfrac{1}{2}\end{matrix}\right.\)

Ta sẽ có hai trường hợp:

TH1: sin x=căn 3/2; cosx=1/2

tan x=sinx/cosx=căn 3

cot x=1/căn 3

TH2: sin x=1/2; cosx=căn 3/2

tan x=sin x/cosx=1/căn 3

cot x=1:1/căn 3=căn 3

1/ \(y'=\left(1-3x\right)'\sqrt{x-3}+\left(1-3x\right)\left(\sqrt{x-3}\right)'=-3\sqrt{x-3}+\dfrac{1}{2\sqrt{x-3}}\left(1-3x\right)\)

2/ \(y'=\dfrac{1}{\sqrt{2x+1}}-\dfrac{1}{\left(x+1\right)^2}\)

3/ \(y'=\dfrac{1}{2}.\sqrt{\dfrac{1+x}{1-x}}.\left(\dfrac{1-x}{1+x}\right)'=\dfrac{1}{2}\sqrt{\dfrac{1+x}{1-x}}.\dfrac{-2}{\left(1+x\right)^2}=-\sqrt{\dfrac{1+x}{1-x}}.\dfrac{1}{\left(1+x\right)^2}\)

4/ \(y'=\left(\cos5x\right)'.\cos7x+\cos5x.\left(\cos7x\right)'=-5\sin5x.\cos7x-7\cos5x\sin7x\)

5/ \(y'=\left(\cos x\right)'\sin^2x+\cos x\left(\sin^2x\right)'=-\sin^3x+2\sin x.\cos^2x\)

6/ \(y'=\left(\tan^42x\right)'=4.\tan^32x.\dfrac{2}{\cos^22x}\)

7/ \(y'=\dfrac{2\sin x+2\cos x-2x.\cos x+2x\sin x}{\left(\sin x+\cos x\right)^2}\)

Ờm, bạn tự rút gọn nhé :) Mình đang hơi lười :b

\(A=a^3-b^3-ab\)

   \(=\left(a-b\right)\left(a^2+ab+b^2\right)-ab\)

   \(=a^2+ab+b^2-ab\) (vì \(a-b=1\))

   \(=a^2+b^2\)

   \(=a^2+\left(a-1\right)^2\)

   \(=2a^2-2a+1\)

  \(=2\left(a^2-a+\frac{1}{4}\right)+\frac{1}{2}\)

  \(=2\left(a-\frac{1}{2}\right)^2+\frac{1}{2}\ge\frac{1}{2}\forall a\)

Dấu "=" xảy ra: \(\Leftrightarrow a-\frac{1}{2}=0\Leftrightarrow a=\frac{1}{2}\)

\(b=a-1=\frac{1}{2}-1=-\frac{1}{2}\)

Vậy \(A_{min}=\frac{1}{2}\Leftrightarrow a=\frac{1}{2},b=-\frac{1}{2}\)

Chúc bạn học tốt.

Đặt \(cosx-sinx=t\Rightarrow-\sqrt{2}\le t\le\sqrt{2}\)

\(t^2=1-2sinx.cosx\Rightarrow sinx.cosx=\dfrac{1-t^2}{2}\)

Pt trở thành:

\(t\left(1+\dfrac{1-t^2}{2}\right)+1=0\)

\(\Leftrightarrow t^3-3t-2=0\)

\(\Leftrightarrow\left(t-2\right)\left(t+1\right)^2=0\Rightarrow\left[{}\begin{matrix}t=2\left(loại\right)\\t=-1\end{matrix}\right.\)

\(\Rightarrow cosx-sinx=-1\)

\(\Leftrightarrow\sqrt[]{2}cos\left(x+\dfrac{\pi}{4}\right)=-1\)

\(\Leftrightarrow cos\left(x+\dfrac{\pi}{4}\right)=-\dfrac{\sqrt{2}}{2}\)

\(\Leftrightarrow cos\left(x+\dfrac{\pi}{4}\right)=cos\left(\dfrac{3\pi}{4}\right)\)

\(\Leftrightarrow...\)

Lời giải:

$m^2=(\sin x+\cos x)^2=\sin ^2x+\cos ^2x+2\sin x\cos x=1+2\sin x\cos x$

$\Rightarrow \sin x\cos x=\frac{m^2-1}{2}$

Ta có:

$|\sin ^3x-\cos ^3x|=|\sin x-\cos x||\sin ^2x+\sin x\cos x+\cos ^2x|$

$=\sqrt{(\sin x-\cos x)^2}|1+\sin x\cos x|$

$=\sqrt{1-2\sin x\cos x}.|1+\sin x\cos x|$

$=\sqrt{1-(m^2-1)}.|1+\frac{m^2-1}{2}|$

$=\sqrt{2-m^2}.\frac{m^2+1}{2}$

\(sinx+cosx=m\\ \Rightarrow sin^2x+cos^2x+2sinx.cosx=m^2\\ \Rightarrow sinx.cosx=\dfrac{1-m^2}{2}\)

Mặt khác: 

\(sinx-cosx=\left(sinx+cosx\right)-2cosx=m-2cosx\)

Có:

\(\left|sin^3x-cos^3x\right|=\left|\left(sinx-cosx\right)\left(sin^2x+sinx.cosx+cos^2x\right)\right|\\ =\left|\left(m-2cosx\right)\left(1+\dfrac{1-m^2}{2}\right)\right|\\ =\left|\left(m-2cosx\right)\left(\dfrac{3-m^2}{2}\right)\right|\)

$\sin x=0,6\\\Leftrightarrow \sin^2 x=0,36\\\Rightarrow \cos^2 x=0,64\\\Leftrightarrow \cos x=0,8(x>0)$