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Bài 2. Đơn giản các biểu thức
a) \(\sin {100^0} + \sin {80^0} + \cos {16^0} + \cos {164^0}\);
b) \(2\sin ({180^0} – \alpha )\cot \alpha – \cos ({180^0} – \alpha )\tan \alpha \cot ({180^0} – \alpha )\) với \({0^0} < \alpha < {90^0}\).
a) Ta có
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\(\eqalign{
& \sin {100^0} = \sin ({180^0} – {80^0}) = \sin {80^0}\,\,\,;\,\,\,\,\cos {164^0} = \cos ({180^0} – {16^0}) = – \cos {16^0} \cr
& \Rightarrow \,\,\,\,\sin {100^0} + \sin {80^0} + \cos {16^0} + \cos {164^0} \cr
& \,\,\,\,\, = \,\sin {80^0} + \sin {80^0} + \cos {16^0} – \cos {16^0} \cr
& \,\,\,\,\, = 2\sin {80^0}. \cr} \)
b) Ta có
\(\eqalign{
& \,\,\,\,2\sin ({180^0} – \alpha )\cot \alpha – \cos ({180^0} – \alpha )\tan \alpha \cot ({180^0} – \alpha ) \cr
& = 2\sin \alpha {{\cos \alpha } \over {\sin \alpha }} – \cos \alpha {{\sin \alpha } \over {\cos \alpha }}{{\cos \alpha } \over {\sin \alpha }} \cr
& = 2\cos \alpha – \cos \alpha \cr
& = \cos \alpha . \cr} \)