a. ĐKXĐ \(x\ne\dfrac{\pi}{4}+\dfrac{k\pi}{2}\)
\(2-cosx=cos2x\)
\(\Leftrightarrow2-cosx=2cos^2x-1\)
\(\Leftrightarrow2cos^2x+cosx-3=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=1\\cosx=-\dfrac{3}{2}\left(loại\right)\end{matrix}\right.\)
\(\Leftrightarrow x=k2\pi\)
c.
ĐKXĐ: \(x\ne\dfrac{\pi}{2}+k2\pi\)
\(cos2x+sinx+sinx-1=0\)
\(\Leftrightarrow1-2sin^2x+2sinx-1=0\)
\(\Leftrightarrow2sinx\left(1-sinx\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\\sinx=1\left(loại\right)\end{matrix}\right.\)
\(\Leftrightarrow x=k\pi\)
b.
ĐKXĐ: \(sinx\ne1\)
\(sin^2x+sinx=-2\left(sinx-1\right)\)
\(\Leftrightarrow sin^2x+3sinx-2=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=\dfrac{-3-\sqrt{17}}{2}\left(loại\right)\\sinx=\dfrac{-3+\sqrt{17}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=arcsin\left(\dfrac{-3+\sqrt{17}}{2}\right)+k2\pi\\x=\pi-arcsin\left(\dfrac{-3+\sqrt{17}}{2}\right)+k2\pi\end{matrix}\right.\)
d.
ĐKXĐ: \(cosx\ne-1\)
\(cos^2x+cosx+2+cosx\left(cosx+1\right)=3\left(cosx+1\right)\)
\(\Leftrightarrow2cos^2x-cosx-1=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=1\\cosx=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k2\pi\\x=\dfrac{2\pi}{3}+k2\pi\\x=-\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\)
a)Đk:\(x\ne\dfrac{\pi}{4}+k\pi\)
Pt \(\Leftrightarrow2-cosx=cos2x\)
\(\Leftrightarrow2-cosx=2cos^2x-1\)
\(\Leftrightarrow2cos^2x+cosx-3=0\)\(\Leftrightarrow\left[{}\begin{matrix}cosx=1\left(tm\right)\\cosx=-\dfrac{3}{2}\left(ktm\right)\end{matrix}\right.\)\(\Rightarrow x=k2\pi;k\in Z\)
b) Đk: \(sinx\ne1\)
Pt \(\Leftrightarrow sin^2x+sinx=-2sinx+2\)
\(\Leftrightarrow sin^2x+3sinx-2=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=\dfrac{-3+\sqrt{17}}{2}\left(tm\right)\\sinx=\dfrac{-3-\sqrt{17}}{2}\left(ktm\right)\end{matrix}\right.\)\(\Rightarrow\left[{}\begin{matrix}x=arc.sin\left(\dfrac{-3+\sqrt{17}}{2}\right)+k2\pi\\x=\pi-arc.sin\left(\dfrac{-3+\sqrt{17}}{2}\right)+k2\pi\end{matrix}\right.\), k nguyên
c)ĐK: \(sinx\ne1\)
Pt \(\Leftrightarrow\dfrac{1-2sin^2x+sinx}{sinx-1}+1=0\)
\(\Leftrightarrow\dfrac{-\left(sinx-1\right)\left(2sinx+1\right)}{sinx-1}+1=0\)
\(\Leftrightarrow-2sinx=0\)\(\Leftrightarrow sinx=0\)(tm) \(\Leftrightarrow x=k\pi\)( k nguyên)
d) Đk: \(cosx\ne-1\)
Pt \(\Leftrightarrow cos^2x+cosx+2+cosx\left(cosx+1\right)=3\left(cosx+1\right)\)
\(\Leftrightarrow2cos^2x-cosx-1=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=1\\cosx=-\dfrac{1}{2}\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=k2\pi\\x=\dfrac{2\pi}{3}+k2\pi\\x=-\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\) (k nguyên)
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