求证tanθ+secθ=tan(θ/2+π/4)

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求证tanθ+secθ=tan(θ/2+π/4)

求证tanθ+secθ=tan(θ/2+π/4)
求证tanθ+secθ=tan(θ/2+π/4)

求证tanθ+secθ=tan(θ/2+π/4)
tanθ+secθ
=sinθ/cosθ+1/cosθ
=(sinθ+1)/cosθ
=[cos(π/2 -θ)+1]/sin(π/2 -θ)
=2cos²(π/4 -θ/2)/[2sin(π/4 -θ/2)cos(π/4 -θ/2)]
=cos(π/4 -θ/2)/sin(π/4 -θ/2)
=cot(π/4 -θ/2)
=tan[π/2-(π/4 -θ/2)]
=tan(θ/2+π/4)

右边=tan(θ/2+π/4)
=(tanθ/2+tanπ/4) / (1-tanθ/2tanπ/4)
=(tanθ/2+1) / (1-tanθ/2)
分子分母同乘以cosθ/2:
=(sinθ/2+cosθ/2) / (cosθ/2-sinθ/2)
分子分母同乘以(cosθ/2+sinθ/2)
=(sinθ/2+cosθ/2)^2 / [(cos...

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右边=tan(θ/2+π/4)
=(tanθ/2+tanπ/4) / (1-tanθ/2tanπ/4)
=(tanθ/2+1) / (1-tanθ/2)
分子分母同乘以cosθ/2:
=(sinθ/2+cosθ/2) / (cosθ/2-sinθ/2)
分子分母同乘以(cosθ/2+sinθ/2)
=(sinθ/2+cosθ/2)^2 / [(cosθ/)^2-(sinθ/2)^2]
=[(sinθ/2)^2+(cosθ/2)^2+2sinθ/2cosθ/2] / cosθ
=(1+sinθ) / cosθ
=(sinθ+1) / cosθ
=sinθ/cosθ + 1/cosθ
=tanθ+secθ=左边,得证。

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