Câu 81 trang 119 Sách bài tập Toán 9 Tập 1: Hãy đơn giản các biểu thức...

Hãy đơn giản các biểu thức:

a) $1 - {\sin ^2}\alpha$;

b) $(1 - \cos \alpha )(1 + \cos \alpha )$;

c) $1 + {\sin ^2}\alpha  + {\cos ^2}\alpha$;

d) $\sin \alpha  - \sin \alpha .{\cos ^2}\alpha$;

e) ${\sin ^4}\alpha  + {\cos ^4}\alpha  + 2.{\sin ^2}\alpha .{\cos ^2}\alpha$;

g) $t{g^2}\alpha  - {\sin ^2}\alpha .t{g^2}\alpha$;

h) ${\cos ^2}\alpha  + t{g^2}\alpha .c{\rm{o}}{{\rm{s}}^2}\alpha$;

i) $t{g^2}\alpha (2.{\cos ^2}\alpha  + {\sin ^2}\alpha  - 1).$

Gợi ý làm bài

a) $1 - {\sin ^2}\alpha  = ({\sin ^2}\alpha  + {\cos ^2}\alpha ) - {\sin ^2}\alpha$

$= {\sin ^2}\alpha  + {\cos ^2}\alpha  - {\sin ^2}\alpha  = {\cos ^2}\alpha$

$\eqalign{ & b)\,(1 - \cos \alpha )(1 + \cos \alpha ) = 1 - {\cos ^2}\alpha \cr & = ({\sin ^2}\alpha + {\cos ^2}\alpha ) - {\cos ^2}\alpha \cr}$

$= {\sin ^2}\alpha  + {\cos ^2}\alpha  - {\cos ^2}\alpha  = {\sin ^2}\alpha$

$\eqalign{ & c)\,1 + {\sin ^2}\alpha + {\cos ^2}\alpha \cr & = 1 + ({\sin ^2}\alpha + {\cos ^2}\alpha ) = 1 + 1 = 2 \cr}$

d) $\sin \alpha  - \sin \alpha .{\cos ^2}\alpha  = \sin \alpha (1 - {\cos ^2}\alpha )$

$= \sin \alpha \left[ {\left( {{{\sin }^2}\alpha  + {{\cos }^2}\alpha } \right) - {{\cos }^2}\alpha } \right]$

$= \sin \alpha ({\sin ^2}\alpha  + {\cos ^2}\alpha  - {\cos ^2}\alpha )$

$= \sin \alpha .{\sin ^2}\alpha  = {\sin ^3}\alpha$

$\eqalign{ & e)\,{\sin ^4}\alpha + {\cos ^4}\alpha + 2.{\sin ^2}\alpha .{\cos ^2}\alpha \cr & = {({\sin ^2}\alpha + {\cos ^2}\alpha )^2} = {1^2} = 1 \cr}$

g) $t{g^2}\alpha  - {\sin ^2}\alpha .t{g^2}\alpha$$= t{g^2}\alpha (1 - {\sin ^2}\alpha )$

$= t{g^2}\left[ {\left( {{{\sin }^2}\alpha  + {{\cos }^2}\alpha } \right) - {{\sin }^2}\alpha } \right]$

$= t{g^2}\alpha .{\cos ^2}\alpha  = {{{{\sin }^2}\alpha } \over {{{\cos }^2}\alpha }}.{\cos ^2}\alpha  = {\sin ^2}\alpha$

$\eqalign{ & h)\,{\cos ^2}\alpha + t{g^2}\alpha .c{\rm{o}}{{\rm{s}}^2}\alpha \cr & = c{\rm{o}}{{\rm{s}}^2}\alpha + {{{{\sin }^2}\alpha } \over {c{\rm{o}}{{\rm{s}}^2}\alpha }}.c{\rm{o}}{{\rm{s}}^2}\alpha \cr & = c{\rm{o}}{{\rm{s}}^2}\alpha + {\sin ^2}\alpha = 1 \cr}$

$\eqalign{ & i)\,t{g^2}\alpha (2.{\cos ^2}\alpha + {\sin ^2}\alpha - 1) \cr & = t{g^2}\alpha .\left[ {{{\cos }^2}\alpha + \left( {{{\cos }^2}\alpha + {{\sin }^2}\alpha } \right) - 1} \right] \cr}$

$= t{g^2}\alpha .({\cos ^2}\alpha  + 1 - 1) = t{g^2}\alpha .{\cos ^2}\alpha$

$= {{{{\sin }^2}\alpha } \over {{{\cos }^2}\alpha }}.{\cos ^2}\alpha  = {\sin ^2}\alpha$