Toán Lớp 11: Giải PT: $sin^{4}$x + $cos^{4}$x = $\frac{-1}{2}$ $cos^{2}$ 2x
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Toán Lớp 11: Giải PT: $sin^{4}$x + $cos^{4}$x = $\frac{-1}{2}$ $cos^{2}$ 2x
Comments ( 1 )
{\sin ^4}x + {\cos ^4}x = – \dfrac{1}{2}{\cos ^2}2x\\
\Leftrightarrow {\left( {{{\sin }^2}x + {{\cos }^2}x} \right)^2} – 2{\sin ^2}x{\cos ^2}x = – \dfrac{1}{2}{\cos ^2}2x\\
\Leftrightarrow 1 – 2{\left( {\dfrac{1}{2}\sin 2x} \right)^2} = – \dfrac{1}{2}{\cos ^2}2x\\
\Leftrightarrow 1 – \dfrac{1}{2}{\sin ^2}2x = – \dfrac{1}{2}{\cos ^2}2x\\
\Leftrightarrow \dfrac{1}{2}{\cos ^2}2x – \dfrac{1}{2}{\sin ^2}2x + 1 = 0\\
\Leftrightarrow \dfrac{1}{2}\left( {{{\cos }^2}2x – {{\sin }^2}2x} \right) = – 1\\
\Leftrightarrow \dfrac{1}{2}\cos 4x = – 1\\
\Leftrightarrow \cos 4x = – 2\left( 1 \right)\\
– 1 \le \cos 4x \le 1 \Rightarrow \left( 1 \right)VN\\
\Rightarrow S = \emptyset
\end{array}$