\(\frac{2.sinx}{cosx}+\frac{cosx}{sinx}-4sinx.cosx-\frac{1}{2sinx.cosx}=0\) (Điều kiện \(x\ne\frac{k\pi}{2}\))

\(\Leftrightarrow\frac{4sin^2x+2cos^2x-8sin^2x.cos^2x-1}{2sinx.cosx}=0\)

\(\Leftrightarrow2sin^2x-8sin^2x.cos^2x+2\left(sin^2x+cos^2x\right)-1=0\)

\(\Leftrightarrow2sin^2x-8sin^2x.\left(1-sin^2x\right)+2-1=0\)

\(\Leftrightarrow8sin^4x-6sin^2x+1=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sin^2x=\frac{1}{4}\\sin^2x=\frac{1}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\frac{1-cos2x}{2}=\frac{1}{4}\\\frac{1-cos2x}{2}=\frac{1}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}cos2x=\frac{1}{2}=cos\frac{\pi}{3}\\cos2x=0=cos\frac{\pi}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=\frac{\pi}{2}+k\pi\\2x=\frac{\pi}{3}+k2\pi\\2x=-\frac{\pi}{3}+k2\pi\end{matrix}\right.\left(k\in Z\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{4}+\frac{k\pi}{4}\\x=\frac{\pi}{6}+k\pi\\x=-\frac{\pi}{6}+k\pi\end{matrix}\right.\left(k\in Z\right)\)

giao với điều kiện \(\Rightarrow\left[{}\begin{matrix}x=\frac{\pi}{4}+\frac{k\pi}{2}\\x=\frac{\pi}{6}+k\pi\\x=-\frac{\pi}{6}+k\pi\end{matrix}\right.\left(k\in Z\right)\)

Đáp án D

a)\(\int \sin ^2\left (\frac{x}{2}\right)dx=\int \frac{1-\cos x }{2}dx=\frac{x}{2}-\frac{\sin x}{2}+c\)

b)\(\int \cos ^2 \left (\frac{x}{2}\right)dx=\int \frac{1+\cos x}{2}dx=\frac{x}{2}+\frac{\sin x}{2}+c\)

c) \(\int \frac{(2x+1)dx}{x^2+x+5}=\int \frac{d(x^2+x+5)}{x^2+x+5}=ln(x^2+x+5)+c\)

d)\(\int (2\tan x+ \cot x)^2dx=4\int \tan ^2 x+\int \cot^2 x+4\int dx=4\int \frac{1-\cos^2 x}{\cos^2 x}dx+\int \frac{1-\sin^2 x}{\sin^2 x}dx+4\int dx \)\( =4\int d(\tan x)-\int d(\cot x)-\int dx=4\tan x-\cot x-x+c\)

\(a,\left(\frac{tan^2x-1}{2tanx}\right)^2-\frac{1}{4sin^2x.cos^2x}=-1\)

\(VT=\left(\frac{tan^2x-1}{2tanx}\right)^2-\frac{1}{4.sin^2x.cos^2x}=\left(\frac{1}{tan2x}\right)^2-\frac{1}{sin^22x}=\left(\frac{cos2x}{sin2x}\right)^2-\frac{1}{sin^22x}=\frac{cos^22x-1}{sin^22x}=\frac{-sin^22x}{sin^22x}=-1=VP\)

b, \(VT=\frac{cos^2x-sin^2x}{sin^4x+cos^4x-sin^2x}=\frac{cos2x}{\left(sin^2x+cos^2x\right)^2-sin^2x-2.sin^2x.cos^2x}=\frac{cos2x}{1-sin^2x-2.sin^2x.cos^2x}=\frac{cos2x}{cos^2x-2.sin^2x.cos^2x}\)

=\(\frac{cos2x}{cos^2x.\left(1-2.sin^2x\right)}=\frac{cos2x}{cos^2x.cos2x}=\frac{1}{cos^2x}=1+tan^2x=VP\)

d, \(VT=\left(\frac{cosx}{1+sinx}+tanx\right).\left(\frac{sinx}{1+cosx}+cotx\right)=\left(\frac{cosx}{1+sinx}+\frac{sinx}{cosx}\right).\left(\frac{sinx}{1+cosx}+\frac{cosx}{sinx}\right)\)

\(=\left(\frac{cos^2x+sinx.\left(1+sinx\right)}{cosx.\left(1+sinx\right)}\right).\left(\frac{sin^2x+cosx.\left(1+cosx\right)}{sinx.\left(1+cosx\right)}\right)=\left(\frac{cos^2x+sinx+sin^2x}{cosx.\left(1+sinx\right)}\right).\left(\frac{sin^2x+cosx+cos^2x}{sinx.\left(1+cosx\right)}\right)\)

=\(\frac{1}{cosx.sinx}=VP\)

e, \(VT=cos^2x.\left(cos^2x+2sin^2x+sin^2x.tan^2x\right)=cos^2x.\left(1+sin^2x.\left(1+tan^2x\right)\right)=cos^2x.\left(1+tan^2x\right)=cos^2x.\frac{1}{cos^2x}=1=VP\)

c, \(VT=\frac{sin^2x}{cosx.\left(1+tanx\right)}-\frac{cos^2x}{sinx.\left(1+cosx\right)}=\frac{sin^3x.\left(1+cosx\right)-cos^3x.\left(1+tanx\right)}{sinx.cosx.\left(1+tanx\right).\left(1+cosx\right)}\)

=\(\frac{sin^3x+sin^3x.cotx-cos^3x-cos^3.tanx}{\left(sinx+cosx\right)^2}=\frac{sin^3x+sin^2xcosx-cos^3x-cos^2sinx}{\left(sinx+cosx\right)^2}=\frac{sin^2x.\left(sinx+cosx\right)-cos^2x.\left(sinx+cosx\right)}{\left(sinx+cosx\right)^2}\)

\(=\frac{\left(sin^2x-cos^2x\right).\left(sinx+cosx\right)}{\left(sinx+cosx\right)^2}=\frac{\left(sinx-cosx\right).\left(sinx+cosx\right).\left(sinx+cosx\right)}{\left(sinx+cosx\right)^2}=sinx-cosx=VP\)

Đây nha bạn

1/ \(sinx=-\frac{1}{2}=sin\left(-\frac{\pi}{6}\right)\)

\(\Rightarrow\left[{}\begin{matrix}x=-\frac{\pi}{6}+k2\pi\\x=\frac{7\pi}{6}+k2\pi\end{matrix}\right.\)

b/ \(cos=-\frac{\sqrt{2}}{2}=cos\left(\frac{3\pi}{4}\right)\)

\(\Rightarrow x=\pm\frac{3\pi}{4}+k2\pi\)

c/ \(tanx=\sqrt{3}=tan\left(\frac{\pi}{3}\right)\)

\(\Rightarrow x=\frac{\pi}{3}+k\pi\)

d/ \(cotx=0\Rightarrow x=\frac{\pi}{2}+k\pi\)

2/

a/ \(sin^2x+sinx-2=0\)

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

\(\Leftrightarrow\left[{}\begin{matrix}sinx=1\\sinx=-2\left(vn\right)\end{matrix}\right.\) \(\Rightarrow x=\frac{\pi}{2}+k2\pi\)

b/ \(cot^2x-2cotx-3=0\)

\(\Leftrightarrow\left(cotx+1\right)\left(cotx-3\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cotx=-1\\cotx=3\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=-\frac{\pi}{4}+k\pi\\x=arccot3+k\pi\end{matrix}\right.\)

3/ \(\Leftrightarrow1-cos2x+1-cos4x+1-cos6x=3\)

\(\Leftrightarrow cos2x+cos6x+cos4x=0\)

\(\Leftrightarrow2coss4x.cos2x+cos4x=0\)

\(\Leftrightarrow cos4x\left(2cos2x+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cos4x=0\\cos2x=-\frac{1}{2}\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}4x=\frac{\pi}{2}+k\pi\\2x=\frac{2\pi}{3}+k2\pi\\2x=-\frac{2\pi}{3}+k2\pi\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=\frac{\pi}{8}+\frac{k\pi}{4}\\x=\frac{\pi}{3}+k\pi\\x=-\frac{\pi}{3}+k\pi\end{matrix}\right.\)