高一数学计算题，急用，在线等

Question

已知sin(x+π/12)²-sin(x-π/12)²=1/4,x∈（π/4，π/2),求tanx.
请写清楚过程，谢谢

Answer 1

[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<x<π/2
π/2<2x<π
所以2x=5π/6
x=5π/12=π/4+π/6
所以tanx=(tanπ/4+tanπ/6)/(1-tanπ/4tanπ/6)=2+√3