Question

giúp mình với ạ Mình cảm ơn

Answer 1

5.

ĐKXĐ: \(x\ne\dfrac{\pi}{2}+k\pi\)

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

\(\Leftrightarrow\dfrac{\left(cosx-sinx\right)}{cosx}\left(sinx+cosx\right)^2=\dfrac{sinx+cosx}{cosx}\)

\(\Leftrightarrow\dfrac{\left(sinx+cosx\right)\left(cos^2x-sin^2x\right)}{cosx}=\dfrac{sinx+cosx}{cosx}\)

\(\Leftrightarrow\dfrac{cos2x\left(sinx+cosx\right)}{cosx}-\dfrac{sinx+cosx}{cosx}=0\)

\(\Leftrightarrow\dfrac{sinx+cosx}{cosx}\left(cos2x-1\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)=0\\cos2x=1\end{matrix}\right.\)

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

Answer 2

6.

\(\Leftrightarrow\dfrac{1}{2}-\dfrac{1}{2}cos6x-\left(\dfrac{1}{2}+\dfrac{1}{2}cos8x\right)=\dfrac{1}{2}-\dfrac{1}{2}cos10x-\left(\dfrac{1}{2}+\dfrac{1}{2}cos12x\right)\)

\(\Leftrightarrow cos6x-cos10x+cos8x-cos12x=0\)

\(\Leftrightarrow2sin2x.sin8x+2sin2x.sin10x=0\)

\(\Leftrightarrow sin2x\left(sin8x+sin10x\right)=0\)

\(\Leftrightarrow2sin2x.sin9x.cosx=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sin2x=0\\sin9x=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{k\pi}{2}\\x=\dfrac{k\pi}{9}\end{matrix}\right.\)

Answer 3

7.

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

\(\Leftrightarrow2sin^3x+2sin^2x+cosx-1=0\)

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

\(\Leftrightarrow2\left(1-cos^2x\right)\left(1+sinx\right)+cosx-1=0\)

\(\Leftrightarrow2\left(1-cosx\right)\left(1+cosx\right)\left(1+sinx\right)-\left(1-cosx\right)=0\)

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

\(\Leftrightarrow\left(1-cosx\right)\left(2sinx.cosx+2\left(sinx+cosx\right)+sin^2x+cos^2x\right)=0\)

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

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

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

\(\Leftrightarrow...\)