已知sin(x+π/12)²-sin(x-π/12)²=1/4,x∈（π/4,π/2),求tanx._数学解答

已知sin(x+π/12)²-sin(x-π/12)²=1/4,x∈（π/4,π/2),求tanx.

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解答

[sin(x+π/12)+sin(x-π/12)][sin(x+π/12)-sin(x-π/12)]=1/4
(sinxcosπ/12+cosxsinπ/12+sinxcosπ/12-cosxsinπ/12)(sinxcosπ/12+cosxsinπ/12-sinxcosπ/12+cosxsinπ/12)=1/4
4sinxcosπ/12*cosxsinπ/12=1/4
(2sinxcosx)(2sinπ/12cosπ/12=1/4
sin2xsinπ/6=1/4
sin2x=1/2
π/4