2.
\(sin3x+cos2x=1+2sinx.cos2x\)
\(\Leftrightarrow sin3x+cos2x=1+sin3x-sinx\)
\(\Leftrightarrow cos2x+sinx-1=0\)
\(\Leftrightarrow-2sin^2x+sinx=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\\sinx=\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=\dfrac{\pi}{6}+k2\pi\\x=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)
1.
\(cos3x-cos4x+cos5x=0\)
\(\Leftrightarrow cos3x+cos5x-cos4x=0\)
\(\Leftrightarrow2cos4x.cosx-cos4x=0\)
\(\Leftrightarrow\left(2cosx-1\right)cos4x=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=\dfrac{1}{2}\\cos4x=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\pm\dfrac{\pi}{3}+k2\pi\\4x=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\pm\dfrac{\pi}{3}+k2\pi\\x=\dfrac{\pi}{8}+\dfrac{k\pi}{4}\end{matrix}\right.\)
3.
\(cos2x-cosx=2sin^2\dfrac{3x}{2}\)
\(\Leftrightarrow2sin\dfrac{3x}{2}.sin\dfrac{x}{2}+2sin^2\dfrac{3x}{2}=0\)
\(\Leftrightarrow2sin\dfrac{3x}{2}.\left(sin\dfrac{x}{2}+sin\dfrac{3x}{2}\right)=0\)
\(\Leftrightarrow sin\dfrac{3x}{2}.sinx.cos\dfrac{x}{2}=0\)
Đến đây dễ rồi tự làm tiếp nha.
4.
\(cos^22x+cos^23x=sin^2x\)
\(\Leftrightarrow1-3sin^2x+cos^23x=0\)
\(\Leftrightarrow3cos^2x-2+4cos^3x-3cosx=0\)
\(\Leftrightarrow\left(cosx+1\right)\left(4cos^2x-cosx-2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=-1\\cosx=\dfrac{1+\sqrt{33}}{8}\\cosx=\dfrac{1-\sqrt{33}}{8}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\pi+k2\pi\\x=\pm arccos\left(\dfrac{1+\sqrt{33}}{8}\right)+k2\pi\\x=\pm arccos\left(\dfrac{1-\sqrt{33}}{8}\right)+k2\pi\end{matrix}\right.\)
5.
\(sin3x.sin5x-cos4x.cos6x=0\)
\(\Leftrightarrow cos2x-cos8x-cos10x-cos2x=0\)
\(\Leftrightarrow cos8x+cos10x=\text{}0\)
\(\Leftrightarrow2cos9x.cosx=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos9x=0\\cosx=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{18}+\dfrac{k\pi}{9}\\x=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
