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Bài 7. Chứng minh các hệ thức sau:
a) \({{1 – 2{{\sin }^2}a} \over {1 + \sin 2a}} = {{1 – \tan a} \over {1 + \tan a}}\)
b) \({{\sin a + \sin 3a + \sin 5a} \over {\cos a + \cos 3a + \cos 5a}} = \tan 3a\)
c) \({{{{\sin }^4}a – {{\cos }^4}a + {{\cos }^2}a} \over {2(1 – \cos a)}} = {\cos ^2}{a \over 2}\)
d) \({{\tan 2x\tan x} \over {\tan 2x – \tan x}} = \sin 2x\)
a)
\(\eqalign{
& {{1 – 2{{\sin }^2}a} \over {1 + \sin 2a}} = {{{{\cos }^2}a – {{\sin }^2}a} \over {{{\cos }^2}a + {{\sin }^2}a + 2\sin a\cos a}} \cr
& = {{\cos a – \sin a} \over {\cos a + \sin a}} = {{1 – {{\sin a} \over {\cos a}}} \over {1 + {{\sin a} \over {\cos a}}}} \cr
& = {{1 – \tan a} \over {1 + \tan a}} \cr} \)
b)
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\(\eqalign{
& {{\sin a + \sin 3a + \sin 5a} \over {\cos a + \cos 3a + \cos 5a}} \cr
& = {{2\sin {{a + 5a} \over 2}\cos {{5a – a} \over 2} + \sin 3a} \over {2\cos {{a + 5a} \over 2}\cos {{5a – a} \over 2} + \cos 3a}} = {{\sin 3a(1 + 2\cos 2a)} \over {\cos 3a(1 + 2\cos 2a)}} \cr
& = \tan 3a \cr} \)
c)
\(\eqalign{
& {{{{\sin }^4}a – {{\cos }^4}a + {{\cos }^2}a} \over {2(1 – \cos a)}} = {{({{\sin }^2}a + {{\cos }^2}a)({{\sin }^2}a – {{\cos }^2}a) + {{\cos }^2}a} \over {2(1 – \cos a)}} \cr
& = {{{{\sin }^2}a – {{\cos }^2}a + {{\cos }^2}a} \over {4{{\sin }^2}{a \over 2}}} = {{4{{\sin }^2}{a \over 2}{{\cos }^2}{a \over 2}} \over {4{{\sin }^2}{a \over 2}}} \cr
& = {\cos ^2}{a \over 2} \cr} \)
d)
\(\eqalign{
& {{\tan 2x\tan x} \over {\tan 2x – \tan x}} \cr
& = {{{{2\tan x} \over {1 – {{\tan }^2}x}}.\tan x} \over {{{2\tan x} \over {1 – {{\tan }^2}x}} – \tan x}} = {{2\tan x} \over {{{\tan }^2}x + 1}} \cr
& = \sin 2x \cr} \)