Bài 7 trang 161 SGK Đại số 10

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) \({{\tan 2x\tan x} \over {\tan 2x - \tan x}} = \sin 2x\)

\(\eqalign{ a) \, \, & {{1 - 2{{\sin }^2}a} \over {1 + \sin 2a}} \cr&= {{{{\cos }^2}a - {{\sin }^2}a} \over {{{\cos }^2}a + {{\sin }^2}a + 2\sin a\cos a}} \cr & = {{\cos a - \sin a} \over {\cos a + \sin a}} = {{1 - {{\sin a} \over {\cos a}}} \over {1 + {{\sin a} \over {\cos a}}}} \cr & = {{1 - \tan a} \over {1 + \tan a}} \cr} \)

\(\eqalign{b) \, \, 
& {{\sin a + \sin 3a + \sin 5a} \over {\cos a + \cos 3a + \cos 5a}} \cr
& = {{2\sin {{a + 5a} \over 2}\cos {{5a - a} \over 2} + \sin 3a} \over {2\cos {{a + 5a} \over 2}\cos {{5a - a} \over 2} + \cos 3a}} \cr&= {{\sin 3a(1 + 2\cos 2a)} \over {\cos 3a(1 + 2\cos 2a)}} \cr
& = \tan 3a \cr} \)

\(\eqalign{d)\, \, 
& {{\tan 2x\tan x} \over {\tan 2x - \tan x}} \cr
& = {{{{2\tan x} \over {1 - {{\tan }^2}x}}.\tan x} \over {{{2\tan x} \over {1 - {{\tan }^2}x}} - \tan x}} \cr&= {{2\tan x} \over {{{\tan }^2}x + 1}} \cr
& = \sin 2x \cr} \)