sin(α+β) - sin(α-β)
= sin(α+β) - sin(α+(-β))
= (sinα cosβ + cosα sinβ) - (sinα cos(-β) + cosα sin(-β))
= (sinα cosβ + cosα sinβ) - (sinα cosβ - cosα sinβ )
= 2 cosα sinβ
0<θ<π/2 のとき、
θ≦tanθ
θ≦sinθ/cosθ
cosθ≦sinθ/θ
sinθ≦θより、sinθ/θ≦1 だから、
cosθ≦sinθ/θ≦1
lim[θ→0]cosθ ≦lim[θ→0](sinθ/θ)≦lim[θ→0] 1
cos(0)≦lim[θ→0](sinθ/θ)≦ 1
1 ≦lim[θ→0](sinθ/θ)≦ 1
lim[θ→0](sinθ/θ)=1
d/dx sin(x)
= lim[ h→0]((sin(x + h) - sin(x))/h)
= lim[ h→0]((sin((x + h/2) + h/2) - sin((x + h/2) - h/2))/h)
= lim[ h→0]((2 cos(x + h/2) sin(h/2))/h)
(∵ sin(α+β) - sin(α-β) = 2 cosα sinβ )
= lim[ h→0]((cos(x + h/2) sin(h/2))/(h/2))
= lim[θ→0]((cos(x + θ) sinθ)/θ )
= lim[θ→0]( cos(x + θ) (sinθ/θ))
= cos(x + 0)・1 (∵ lim[θ→0](sinθ/θ)=1 )
= cos(x)
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