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Chứng minh các đẳng thức
a) \({{\tan \alpha – \tan \beta } \over {{\rm{cot}}\beta {\rm{ – cot}}\alpha }} = \tan \alpha \tan \beta\)
b) \(\tan {100^0} + {{\sin {{530}^0}} \over {1 + \sin {{640}^0}}} = {1 \over {\sin {{10}^0}}}\)
c) \(2({\sin ^6}\alpha + c{\rm{o}}{{\rm{s}}^6}\alpha ) + 1 = 3({\sin ^4}\alpha + c{\rm{o}}{{\rm{s}}^4}\alpha )\)
Gợi ý làm bài
a)
\(\eqalign{
& {{\tan \alpha – \tan \beta } \over {{\rm{cot}}\beta {\rm{ – cot}}\alpha }} = {{\tan \alpha – \tan \beta } \over {{1 \over {\tan \beta }} – {1 \over {\tan \alpha }}}} \cr
& = {{\tan \alpha – \tan \beta } \over {{{\tan \alpha – \tan \beta } \over {tan\alpha \tan \beta }}}} = \tan \alpha \tan \beta \cr} \)
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b)
\(\eqalign{
& \tan {100^0} + {{\sin {{530}^0}} \over {1 + \sin {{640}^0}}} \cr
& = \tan ({90^0} + {10^0}) + {{\sin ({{360}^0} + {{170}^0})} \over {1 + \sin ({{720}^0} – {{80}^0})}} \cr} \)
\(\eqalign{
& = – \cot {10^0} + {{\sin {{170}^0}} \over {1 – \sin {{80}^0}}} \cr
& = – {{\cos {{10}^0}} \over {\sin {{10}^0}}} + {{\sin {{10}^0}} \over {1 – c{\rm{os1}}{{\rm{0}}^0}}} \cr} \)
\( = {{ – \cos {{10}^0} + {{\cos }^2}{{10}^0} + {{\sin }^2}{{10}^0}} \over {\sin {{10}^0}(1 – c{\rm{os1}}{{\rm{0}}^0})}} = {1 \over {\sin {{10}^0}}}\)
\(\eqalign{
& c)2({\sin ^6}\alpha + c{\rm{o}}{{\rm{s}}^6}\alpha ) + 1 \cr
& = 2({\sin ^2}x + c{\rm{o}}{{\rm{s}}^2}x)({\sin ^4}x – {\sin ^2}x{\cos ^2}x + c{\rm{o}}{{\rm{s}}^4}x) + 1 \cr
& = 2({\sin ^4}x + c{\rm{o}}{{\rm{s}}^4}x) + {({\sin ^2}x + c{\rm{o}}{{\rm{s}}^2}x)^2} – 2{\sin ^{^2}}x{\cos ^2}x \cr
& = 2({\sin ^4}x + c{\rm{o}}{{\rm{s}}^4}x) + ({\sin ^4}x + c{\rm{o}}{{\rm{s}}^4}x) \cr
& = 3({\sin ^4}\alpha + c{\rm{o}}{{\rm{s}}^4}\alpha ) \cr} \)