Question

求值:cosπ/7cos2π/7cos3π/7cos4π/7cos5π/7cos6π/7

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Answer 1

解：由诱导公式可得：cos6π/7=-cosπ/7；
cosπ/7=-cos6π/7,cos3π/7=-cos4π/7,cos5π/7=-cos2π/7
∴ cos2π/7cos4π/7cos6π/7
=-cosπ/7cos2π/7cos4π/7
=-sinπ/7cosπ/7cos2π/7cos4π/7/sinπ/7
=-1/2·sin2π/7cos2π/7cos4π/7/sinπ/7
=-1/4·sin4π/7cos4π/7/sinπ/7
=-1/8·sin8π/7/sinπ/7
=-1/8·(-sinπ/7)/sinπ/7
=1/8
而cosπ/7cos3π/7cos5π/7=(-cos6π/7)(-cos4π/7)(-cos2π/7)
=-cos2π/7cos4π/7cos6π/7=-1/8
∴原式=-1/64