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Không dùng bảng số và máy tính, rút gọn các biểu thức
a) \(A = \tan {18^0}\tan {288^0} + \sin {32^0}\sin {148^0} – \sin {302^0}\sin {122^0}\)
b) \(B = {{1 + {{\sin }^4}\alpha – c{\rm{o}}{{\rm{s}}^4}\alpha } \over {1 – {{\sin }^6}\alpha – c{\rm{o}}{{\rm{s}}^6}\alpha }}\)
Gợi ý làm bài
a)
\(A = \tan ({90^0} – {72^0})\tan ({360^0} – {72^0}) + \sin {32^0}\sin ({180^0} – {32^0}) – \sin ({360^0} – {58^0})\sin ({180^0} – {58^0})\)
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\(\eqalign{
& \cot {72^0}( – \tan {72^0}) + {\sin ^2}{32^0} + {\sin ^2}{58^0} \cr
& = – 1 + {\sin ^2}{32^0} + c{\rm{o}}{{\rm{s}}^2}{32^0} \cr
& = – 1 + 1 = 0 \cr} \)
b)
\(\eqalign{
& B = {{1 + ({{\sin }^2}\alpha + c{\rm{o}}{{\rm{s}}^2}\alpha )(si{n^2}\alpha – c{\rm{o}}{{\rm{s}}^2}\alpha )} \over {1 – ({{\sin }^2}\alpha + c{\rm{o}}{{\rm{s}}^2}\alpha )({{\sin }^4}\alpha – {{\sin }^2}\alpha c{\rm{o}}{{\rm{s}}^2}\alpha + c{\rm{o}}{{\rm{s}}^4}\alpha )}} \cr
& = {{1 + {{\sin }^2}\alpha – c{\rm{o}}{{\rm{s}}^2}\alpha } \over {1 – {\rm{[}}{{({{\sin }^2}\alpha + c{\rm{o}}{{\rm{s}}^2}\alpha )}^2} – 3{{\sin }^2}\alpha c{\rm{o}}{{\rm{s}}^2}\alpha }} \cr
& = {{3{{\sin }^2}\alpha } \over {3{{\sin }^2}\alpha c{\rm{o}}{{\rm{s}}^2}\alpha }} = {2 \over 3}(1 + {\tan ^2}\alpha ) \cr} \)