Câu 81 trang 119 Sách Bài Tập (SBT) Toán 9 Tập 1

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

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

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

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

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

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

\( = {{{{\sin }^2}\alpha } \over {{{\cos }^2}\alpha }}.{\cos ^2}\alpha  = {\sin ^2}\alpha \)

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