11.\(cos\dfrac{x}{2}=-cos\left(2x-30^0\right)\)
\(\Leftrightarrow cos\dfrac{x}{2}=-cos\left(2x-\dfrac{\pi}{6}\right)\)
\(\Leftrightarrow cos\dfrac{x}{2}=cos\left(2x+\dfrac{5\pi}{6}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{x}{2}=2x+\dfrac{5\pi}{6}+k2\pi\\\dfrac{x}{2}=-2x-\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)(\(k\in Z\))\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{5\pi}{9}-\dfrac{k4\pi}{3}\\x=-\dfrac{\pi}{3}+\dfrac{k4\pi}{5}\end{matrix}\right.\)(\(k\in Z\))
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12) \(sinx-cos2x=0\Leftrightarrow sinx-1+2sin^2x=0\)
\(\Leftrightarrow\left(2sinx-1\right)\left(sinx+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=\dfrac{1}{2}\\sinx=-1\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}sinx=sin\dfrac{\pi}{6}\\sinx=-1\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+k2\pi\\x=\dfrac{5\pi}{6}+k2\pi\\x=\dfrac{-\pi}{2}+k2\pi\end{matrix}\right.\)(\(k\in Z\))
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14)\(cos2x.cos4x=cos3x.cos5x\)
\(\Leftrightarrow\dfrac{1}{2}\left(cos2x+cos6x\right)=\dfrac{1}{2}\left(cos2x+cos8x\right)\)
\(\Leftrightarrow cos6x=cos8x\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-k\pi\\x=\dfrac{k\pi}{7}\end{matrix}\right.\)(\(k\in Z\))
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15. \(cosx+sinx=\sqrt{2}\)
\(\Leftrightarrow\dfrac{1}{\sqrt{2}}.cosx+\dfrac{1}{\sqrt{2}}.sinx=1\)
\(\Leftrightarrow sin\left(\dfrac{\pi}{4}+x\right)=1\)
\(\Leftrightarrow x=\dfrac{\pi}{4}+k2\pi\) (\(k\in Z\))
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