tan(X/2+π/4)+tan(x/2-π/4)=2tanx?难道是题目出错了?右边题目是2tanx 亲的答案是tan2x?

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tan(X/2+π/4)+tan(x/2-π/4)=2tanx?难道是题目出错了?右边题目是2tanx 亲的答案是tan2x?
tan(X/2+π/4)+tan(x/2-π/4)=2tanx?
难道是题目出错了?
右边题目是2tanx
亲的答案是tan2x?

tan(X/2+π/4)+tan(x/2-π/4)=2tanx?难道是题目出错了?右边题目是2tanx 亲的答案是tan2x?
tan(X/2+π/4)+tan(x/2-π/4)=(tanx/2+1)/(1-tanx/2)+(tanx/2-1)/(1+tanx/2)=[(tanx/2+1)^2-(tanx/2-1)^2]/[(tanx/2+1)(1-tanx/2)]
=4tanx/2/(1-tanx/2^2)
=2(2tanx/2)/(1-tanx/2^2)
=2tanx

tanπ/4=1
所以tan(x/2+π/4)=(tanx/2+1)/(1-tanx/2)
tan(x/2-π/4)=(tana/2-1)/(1+tanx/2)
为了方便,令a=tanx/2
所以左边=(a+1)/(1-a)+(a-1)/(1+a)
=[(a+1)^2-(1-a)^2]/(1+a)(1-a)
=2a/(1-a^2)
=2tan(x/2)/[1-tan²(x/2)]
=tan2x=右边
命题得证

= = 觉得好眼熟啊 高中毕业太久了 俺忘了