已知tan(α+β)=2/5,tan(α+π/4)=3/22,那么tan(β-π/4)的值是?RT,要详解.

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已知tan(α+β)=2/5,tan(α+π/4)=3/22,那么tan(β-π/4)的值是?RT,要详解.
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已知tan(α+β)=2/5,tan(α+π/4)=3/22,那么tan(β-π/4)的值是?RT,要详解.
已知tan(α+β)=2/5,tan(α+π/4)=3/22,那么tan(β-π/4)的值是?
RT,要详解.

已知tan(α+β)=2/5,tan(α+π/4)=3/22,那么tan(β-π/4)的值是?RT,要详解.
tan(α+π/4)
=tan[α+β-(β-π/4)]
=[tan(α+β)-tan(β-π/4)]/[1+tan(α+β)tan(β-π/4)]
3/22=[2/5-tan(β-π/4)]/[1+2/5tan(β-π/4)]
3/22*[1+2/5tan(β-π/4)]=[2/5-tan(β-π/4)]
3/22+3/55*tan(β-π/4)=2/5-tan(β-π/4)
3/55*tan(β-π/4)+tan(β-π/4)=2/5-3/22
58/55*tan(β-π/4)=29/110
58*tan(β-π/4)=29/2
tan(β-π/4)=29/58*2=1/4

tan(β-π/4)=tan(α+β-α-π/4)
=[tan(α+β)-tan(α+π/4)]/[1+tan(α+β)tan(α+π/4)]
=(2/5-3/22)/(1+3/55)
=29/116
=1/4