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Bài 8. Rút gọn các biểu thức sau:
a) \({{1 + \sin 4a – \cos 4a} \over {1 + \cos 4a + \sin 4a}}\)
b) \({{1 + \cos a} \over {1 – \cos a}}{\tan ^2}{a \over 2} – {\cos ^2}a\)
c) \({{\cos 2x – \sin 4x – \cos 6x} \over {\cos 2x + \sin 4x – \cos 6x}}\)
a)
\(\eqalign{
& {{1 + \sin 4a – \cos 4a} \over {1 + \cos 4a + \sin 4a}} = {{2{{\sin }^2}2a + 2\sin 2a\cos 2a} \over {2{{\cos }^2}2a + 2\sin 2a\cos 2a}} \cr
& = {{2\sin 2a(\sin 2a + \cos 2a)} \over {2\cos 2a(\sin 2a + \cos 2a)}} = \tan 2a \cr} \)
b)
\(\eqalign{
& {{1 + \cos a} \over {1 – \cos a}}{\tan ^2}{a \over 2} – {\cos ^2}a = {{2{{\cos }^2}{a \over 2}} \over {2{{\sin }^2}{a \over 2}}}.{{2{{\sin }^2}{a \over 2}} \over {2{{\cos }^2}{a \over 2}}} – {\cos ^2}{a \over 2} \cr
& = 1 – {\cos ^2}{a \over 2} = {\sin ^2}{a \over 2} \cr} \)
c)
\(\eqalign{
& {{\cos 2x – \sin 4x – \cos 6x} \over {\cos 2x + \sin 4x – \cos 6x}} = {{(cos2x – \cos 6x) – sin4x} \over {(cos2x – \cos 6x) + sin4x}} \cr
& = {{-2\sin {{2x + 6x} \over 2}\sin {{6x – 2x} \over 2} – \sin 4x} \over {-2\sin {{2x + 6x} \over 2}\sin {{2x – 6x} \over 2} + \sin 4x}} \cr
& = {{2\sin 2x – 1} \over {2\sin 2x + 1}} \cr} \)