Question

证明：tan3/2X－tan1/2x=2sinx/（cosx＋cos2x）.我会证明,只是我只能找到一种解法,哪位哥哥姐

Answer 1

证明：tan(3/2)X-tan(1/2)x=2sinx/（cosx＋cos2x）.

证明一：左边={sin[(3/2)x]/cos(3/2)x}-{sin[(1/2)x]/cos(1/2)x}

={sin[(3/2)xcos(1/2)x-cos[(3/2)x]sin(1/2)x}/[cos(3x/2)cos(x/2)

=[sin(3x/2-x/2)]/{(1/2)[cos(3x/2+x/2)+cos(3x/2-x/2)]}

=2(sinx)/(cos2x+cosx)=右边

证明二：右边=[sin(3x/2-x/2)]/{(1/2)[cos(3x/2+x/2)+cos(3x/2-x/2)]}

={sin[(3/2)xcos(1/2)x-cos[(3/2)x]sin(1/2)x}/[cos(3x/2)cos(x/2)

={sin[(3/2)x]/cos(3/2)x}-{sin[(1/2)x]/cos(1/2)x}

=tan(3x/2)-tan(x/2)=左边.