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Câu hỏi: 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) \(ta{n^2}\alpha - {\sin ^2}\alpha .ta{n^2}\alpha \);
h) \({\cos ^2}\alpha + ta{n^2}\alpha .c{\rm{o}}{{\rm{s}}^2}\alpha \);
i) \(ta{n^2}\alpha (2.{\cos ^2}\alpha + {\sin ^2}\alpha - 1).\)
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) \(ta{n^2}\alpha - {\sin ^2}\alpha .ta{n^2}\alpha \);
h) \({\cos ^2}\alpha + ta{n^2}\alpha .c{\rm{o}}{{\rm{s}}^2}\alpha \);
i) \(ta{n^2}\alpha (2.{\cos ^2}\alpha + {\sin ^2}\alpha - 1).\)
Phương pháp giải
Áp dụng các kiến thức:
1) \({\sin ^2}\alpha + {\cos ^2}\alpha =1\)
2) \(ta{n^2}\alpha = \displaystyle {{{{\sin }^2}\alpha } \over {{{\cos }^2}\alpha }}\)
Lời giải chi tiết
a) \(1 - {\sin ^2}\alpha = ({\sin ^2}\alpha + {\cos ^2}\alpha ) - {\sin ^2}\alpha \)
\( = {\sin ^2}\alpha + {\cos ^2}\alpha - {\sin ^2}\alpha = {\cos ^2}\alpha \)
b)
\(\eqalign{
&(1 - \cos \alpha )(1 + \cos \alpha ) = 1 - {\cos ^2}\alpha \cr
& = ({\sin ^2}\alpha + {\cos ^2}\alpha ) - {\cos ^2}\alpha \cr} \)
\( = {\sin ^2}\alpha + {\cos ^2}\alpha - {\cos ^2}\alpha = {\sin ^2}\alpha \)
c)
\(\eqalign{
& 1 + {\sin ^2}\alpha + {\cos ^2}\alpha \cr
& = 1 + ({\sin ^2}\alpha + {\cos ^2}\alpha ) = 1 + 1 = 2 \cr} \)
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) \(ta{n^2}\alpha - {\sin ^2}\alpha .ta{n^2}\alpha \)\( = ta{n^2}\alpha (1 - {\sin ^2}\alpha )\)
\( = ta{n^2}\left[ {\left( {{{\sin }^2}\alpha + {{\cos }^2}\alpha } \right) - {{\sin }^2}\alpha } \right]\)
\( = ta{n^2}\alpha .{\cos ^2}\alpha = \displaystyle {{{{\sin }^2}\alpha } \over {{{\cos }^2}\alpha }}.{\cos ^2}\alpha\)\( = {\sin ^2}\alpha \)
\(\eqalign{
& h) {\cos ^2}\alpha + ta{n^2}\alpha .c{\rm{o}}{{\rm{s}}^2}\alpha \cr
& = c{\rm{o}}{{\rm{s}}^2}\alpha + {{{{\sin }^2}\alpha } \over {c{\rm{o}}{{\rm{s}}^2}\alpha }}.c{\rm{o}}{{\rm{s}}^2}\alpha \cr
& = c{\rm{o}}{{\rm{s}}^2}\alpha + {\sin ^2}\alpha = 1 \cr} \)
i)
\( ta{n^2}\alpha (2.{\cos ^2}\alpha + {\sin ^2}\alpha - 1) \)
\( = ta{n^2}\alpha .\)\(\left[ {{{\cos }^2}\alpha + \left( {{{\cos }^2}\alpha + {{\sin }^2}\alpha } \right) - 1} \right] \)
\( = ta{n^2}\alpha .({\cos ^2}\alpha + 1 - 1)\)\( = ta{n^2}\alpha .{\cos ^2}\alpha \)
\( = \displaystyle {{{{\sin }^2}\alpha } \over {{{\cos }^2}\alpha }}.{\cos ^2}\alpha = {\sin ^2}\alpha \)
Áp dụng các kiến thức:
1) \({\sin ^2}\alpha + {\cos ^2}\alpha =1\)
2) \(ta{n^2}\alpha = \displaystyle {{{{\sin }^2}\alpha } \over {{{\cos }^2}\alpha }}\)
Lời giải chi tiết
a) \(1 - {\sin ^2}\alpha = ({\sin ^2}\alpha + {\cos ^2}\alpha ) - {\sin ^2}\alpha \)
\( = {\sin ^2}\alpha + {\cos ^2}\alpha - {\sin ^2}\alpha = {\cos ^2}\alpha \)
b)
\(\eqalign{
&(1 - \cos \alpha )(1 + \cos \alpha ) = 1 - {\cos ^2}\alpha \cr
& = ({\sin ^2}\alpha + {\cos ^2}\alpha ) - {\cos ^2}\alpha \cr} \)
\( = {\sin ^2}\alpha + {\cos ^2}\alpha - {\cos ^2}\alpha = {\sin ^2}\alpha \)
c)
\(\eqalign{
& 1 + {\sin ^2}\alpha + {\cos ^2}\alpha \cr
& = 1 + ({\sin ^2}\alpha + {\cos ^2}\alpha ) = 1 + 1 = 2 \cr} \)
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) \(ta{n^2}\alpha - {\sin ^2}\alpha .ta{n^2}\alpha \)\( = ta{n^2}\alpha (1 - {\sin ^2}\alpha )\)
\( = ta{n^2}\left[ {\left( {{{\sin }^2}\alpha + {{\cos }^2}\alpha } \right) - {{\sin }^2}\alpha } \right]\)
\( = ta{n^2}\alpha .{\cos ^2}\alpha = \displaystyle {{{{\sin }^2}\alpha } \over {{{\cos }^2}\alpha }}.{\cos ^2}\alpha\)\( = {\sin ^2}\alpha \)
\(\eqalign{
& h) {\cos ^2}\alpha + ta{n^2}\alpha .c{\rm{o}}{{\rm{s}}^2}\alpha \cr
& = c{\rm{o}}{{\rm{s}}^2}\alpha + {{{{\sin }^2}\alpha } \over {c{\rm{o}}{{\rm{s}}^2}\alpha }}.c{\rm{o}}{{\rm{s}}^2}\alpha \cr
& = c{\rm{o}}{{\rm{s}}^2}\alpha + {\sin ^2}\alpha = 1 \cr} \)
i)
\( ta{n^2}\alpha (2.{\cos ^2}\alpha + {\sin ^2}\alpha - 1) \)
\( = ta{n^2}\alpha .\)\(\left[ {{{\cos }^2}\alpha + \left( {{{\cos }^2}\alpha + {{\sin }^2}\alpha } \right) - 1} \right] \)
\( = ta{n^2}\alpha .({\cos ^2}\alpha + 1 - 1)\)\( = ta{n^2}\alpha .{\cos ^2}\alpha \)
\( = \displaystyle {{{{\sin }^2}\alpha } \over {{{\cos }^2}\alpha }}.{\cos ^2}\alpha = {\sin ^2}\alpha \)