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Giải các phương trình :
a. \({\sin ^4}x + {\cos ^4}x = {3 \over 4}\)
b. \({\sin ^2}2x – {\sin ^2}x = {\sin ^2}{\pi \over 4}\)
c. \(\cos x\cos 2x = \cos 3x\)
d. \(\tan 2x – \sin 2x + \cos 2x – 1 = 0\)
a.
\(\eqalign{ & {\sin ^4}x + {\cos ^4}x = {3 \over 4} \cr & \Leftrightarrow 1 – 2{\sin ^2}x{\cos ^2}x = {3 \over 4} \cr & \Leftrightarrow 1 – {1 \over 2}{\sin ^2}2x = {3 \over 4} \cr & \Leftrightarrow {\sin ^2}2x = {1 \over 2} \Leftrightarrow {{1 – \cos 4x} \over 2} = {1 \over 2} \cr & \Leftrightarrow \cos 4x = 0 \Leftrightarrow x = {\pi \over 8} + k{\pi \over 4},k \in\mathbb Z \cr} \)
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b.
\(\eqalign{ & {\sin ^2}2x – {\sin ^2}x = {\sin ^2}{\pi \over 4} \cr & \Leftrightarrow 4{\sin ^2}x{\cos ^2}x – {\sin ^2}x = {1 \over 2} \cr & \Leftrightarrow 8{\sin ^2}x\left( {1 – {{\sin }^2}x} \right) – 2{\sin ^2}x = 1 \cr & \Leftrightarrow 8{\sin ^4}x – 6{\sin ^2}x + 1 = 0 \cr & \Leftrightarrow \left[ {\matrix{ {{{\sin }^2}x = {1 \over 2}} \cr {{{\sin }^2}x = {1 \over 4}} \cr } } \right. \Leftrightarrow \left[ {\matrix{ {{{1 – \cos 2x} \over 2} = {1 \over 2}} \cr {{{1 – \cos 2x} \over 2} = {1 \over 4}} \cr } } \right. \cr & \Leftrightarrow \left[ {\matrix{ {\cos 2x = 0} \cr {\cos 2x = {1 \over 2}} \cr } } \right. \Leftrightarrow \left[ {\matrix{ {x = {\pi \over 4} + k{\pi \over 2}} \cr {x = {\pi \over 6} + k{\pi \over 2}} \cr } } \right. \cr} \)
c.
\(\eqalign{ & \cos x\cos 2x = \cos 3x \cr & \Leftrightarrow {1 \over 2}\left( {\cos 3x + \cos x} \right) = \cos 3x \cr & \Leftrightarrow \cos 3x = \cos x \cr & \Leftrightarrow \left[ {\matrix{ {3x = x + k2\pi } \cr {3x = – x + k2\pi } \cr } } \right. \Leftrightarrow \left[ {\matrix{ {x = k\pi } \cr {x = k{\pi \over 2}} \cr } } \right.\cr& \Leftrightarrow x = k{\pi \over 2},k \in\mathbb Z \cr} \)
d. Điều kiên: \(\cos 2x \ne0\)
\(\eqalign{ & \tan 2x – \sin 2x + \cos 2x – 1 = 0 \cr & \Leftrightarrow \tan 2x\left( {1 – \cos 2x} \right) – \left( {1 – \cos 2x} \right) = 0 \cr & \Leftrightarrow \left( {1 – \cos 2x} \right)\left( {\tan 2x – 1} \right) = 0 \cr & \Leftrightarrow \left[ {\matrix{ {\tan 2x = 1} \cr {\cos 2x = 1} \cr } } \right. \Leftrightarrow \left[ {\matrix{ {x = {\pi \over 8} + k{\pi \over 2}} \cr {x = k\pi } \cr } } \right. k \in\mathbb Z \cr} \)