# Câu 4 trang 223 SGK Đại số và Giải tích 11 Nâng cao

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. $$\tan 2x - \sin 2x + \cos 2x - 1 = 0$$

Giải:

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}

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}