Đáp án B

Đáp án A

\(sin2x-cos2x+3sinx-cosx-1=0\)

\(\Leftrightarrow2sinxcosx-\left(1-2sin^2x\right)+3sinx-cosx-1=0\)

\(\Leftrightarrow2sinxcosx-1+2sin^2x+3sinx-cosx-1=0\)

\(\Leftrightarrow2sin^2x+3sinx-2+cosx\left(2sinx-1\right)=0\)

\(\Leftrightarrow2\left(sinx-\dfrac{1}{2}\right)\left(sinx+2\right)+cosx\left(2sinx-1\right)=0\)

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

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

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

\(\Leftrightarrow\left[{}\begin{matrix}sinx=\dfrac{1}{2}\\sinx+cosx=-2\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}sinx=sin\dfrac{\pi}{6}\\\sqrt[]{2}\left(sinx.\dfrac{1}{\sqrt[]{2}}+cosx.\dfrac{1}{\sqrt[]{2}}\right)=-2\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+k2\pi\\x=\dfrac{5\pi}{6}+k2\pi\\\sqrt[]{2}sin\left(x+\dfrac{\pi}{4}\right)=-2\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+k2\pi\\x=\dfrac{5\pi}{6}+k2\pi\\sin\left(x+\dfrac{\pi}{4}\right)=-\sqrt[]{2}\left(vô.lý\right)\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+k2\pi\\x=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\) \(\left(k\in Z\right)\)

a) <=> 4sinxcosx -(2cos²x-1)=7sinx+2cosx-4

<=> 2cos²x+(2-4sinx)cosx+7sinx-5=0

- sinx=1 => 2cos²x-2cosx+2=0 

pt trên vn

b) <=> 2sinxcosx-1+2sin²x+3sinx-cosx-1=0

<=> cos(2sinx-1)+2sin²x+3sinx-2=0

<=> cosx(2sinx-1)+(2sinx-1)(sinx+2)=0

<=> (2sinx-1)(cosx+sinx+2)=0

<=> sinx=1/2 hoặc cosx+sinx=-2(vn)

<=> x= \(\frac{\pi}{6}+k2\pi\) hoặc \(x=\frac{5\pi}{6}+k2\pi\left(k\in Z\right)\)

ĐKXĐ: \(cosx\ne-\dfrac{\sqrt{3}}{2}\) \(\Rightarrow\left[{}\begin{matrix}x\ne\dfrac{5\pi}{6}+k2\pi\\x\ne\dfrac{7\pi}{6}+k2\pi\end{matrix}\right.\)

\(pt\Rightarrow3-\left(1-2sin^2x\right)+2sinx.cosx-5sinx-cosx=0\)

\(\Leftrightarrow2sin^2x-5sinx+2+cosx\left(2sinx-1\right)=0\)

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

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

\(\Rightarrow\left[{}\begin{matrix}sinx=\dfrac{1}{2}\\sinx+cosx=2\left(vn\right)\end{matrix}\right.\) 

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

Loại nghiệm

\(\Rightarrow x=\dfrac{\pi}{6}+k2\pi\)

\(0\le\dfrac{\pi}{6}+k2\pi\le2022\pi\Rightarrow0\le k\le1010\)

\(\Rightarrow\sum x=1011.\dfrac{\pi}{6}+2\pi\left(0+1+2+...+1010\right)=\dfrac{1011\pi}{6}+2\pi.\dfrac{1010.1011}{2}=...\)

=sin2x-1

Hướng dẫn giải:

Chọn A.

 không là nghiệm của phương trình

Chia 2 vế phương trình cho cos²x ta được  - 3tanx + tan²x = 2(1+ tan²x)

Đáp án D