已知sin(x-3π/4)cos(x-π/4)=-1/4,求cos4x.

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已知sin(x-3π/4)cos(x-π/4)=-1/4,求cos4x.
已知sin(x-3π/4)cos(x-π/4)=-1/4,求cos4x.

已知sin(x-3π/4)cos(x-π/4)=-1/4,求cos4x.
∵sin(x-3π/4)cos(x-π/4)=-1/4
==>[sin((x-3π/4)+(x-π/4))+sin((x-3π/4)-(x-π/4))]/2=-1/4 (应用正弦积化和差公式)
==>sin(2x-π)+sin(-π/2)=-1/2
==>-sin(2x)-1=-1/2
==>sin(2x)=-1/2
∴cos(4x)=1-2sin²(2x) (应用余弦倍角公式)
=1-2(-1/2)²
=1/2