Bài 21 trang 218 Sách bài tập (SBT) Toán Đại số 10: Rút gọn...

Rút gọn

a) ${{{{\sin }^2}2\alpha  + 4{{\sin }^4}\alpha  - 4{{\sin }^2}\alpha c{\rm{o}}{{\rm{s}}^2}\alpha } \over {4 - {{\sin }^2}2\alpha  - 4{{\sin }^2}\alpha }}$

b) $3 - 4\cos 2a + \cos 4a$

c) $\cos 4a - \sin 4a\cot 2a$

d) ${{{\mathop{\rm cota}\nolimits}  + \tan a} \over {1 + \tan 2a\tan a}}$

Gợi ý làm bài

a) 

$\eqalign{ & {{{{\sin }^2}2\alpha + 4{{\sin }^2}4\alpha - 4{{\sin }^2}\alpha {{\cos }^2}\alpha } \over {4 - {{\sin }^2}2\alpha - 4{{\sin }^2}\alpha }} \cr & = {{{{\sin }^2}2\alpha + 4{{\sin }^4}\alpha - {{\sin }^2}2\alpha } \over {4{{\cos }^2}a - 4{{\sin }^2}2\alpha {{\cos }^2}\alpha }} \cr}$

$= {{4{{\sin }^2}\alpha } \over {4co{s^2}\alpha (1 - {{\sin }^2}\alpha )}} = {\tan ^4}\alpha$

b) 

$\eqalign{ & 3 - 4\cos 2a + \cos 4a \cr & = 3 - 4(1 - 2{\sin ^2}a) + (1 - 2{\sin ^2}2a) \cr}$

$\eqalign{ & = 8{\sin ^2}a - 8{\sin ^2}a{\cos ^2}a \cr & = 8{\sin ^2}a(1 - {\cos ^2}a) \cr}$

$= 8{\sin ^4}a$

c) 

$\eqalign{ & \cos 4a - \sin 4a\cot 2a \cr & = 2{\cos ^2}2a - 1 - 2\sin 2a\cos 2a{{\cos 2a} \over {\sin 2a}} = - 1 \cr}$

d) ${{\cot a + \tan a} \over {1 + \tan 2a\tan a}} = {{{{\cos a} \over {\sin a}} + {{\sin a} \over {\cos a}}} \over {1 + {{\sin 2a\sin a} \over {\cos 2a\cos a}}}}$

$= {1 \over {\sin a\cos a}}.{{\cos acos2a} \over {\cos 2a\cos a + \sin 2a\sin a}}$

$= {2 \over {\sin 2a}}.{{\cos acos2a} \over {\cos (2a - a)}} = 2\cot 2a$