(tan45+x)-tan(45-x)/tan(45+x)+tan(45-x)
=[ (tan45°＋tanx)/(1-tan45°tanx) -(tan45°-tanx)/(1+tan45°tanx)]/[ (tan45°＋tanx)/(1-tan45°tanx) -(tan45°-tanx)/(1+tan45°tanx)]
=[(1+tanx)/(1-tanx)-(1-tanx)/(1+tanx)]/[(1+tanx)/(1-tanx)+(1-tanx)/(1+tanx)]
= [(1+tanx)²-(1-tanx)²]/ [(1+tanx)²+(1-tanx)²]
=[4tanx]/[2+2tan²x]
=2tanx/(1+tan²x)
=sin2x