求证:tan^2(α)-1/ tan^2(α)=-2sin4α/sin^3(2α)

求证:tan^2(α)-1/ tan^2(α)=-2sin4α/sin^3(2α)
求证:tan^2(α)-1/ tan^2(α)=-2sin4α/sin^3(2α)

求证:tan^2(α)-1/ tan^2(α)=-2sin4α/sin^3(2α)
左边=tan^2(α)-1/ tan^2(α)
= sin^2(α)/cos^2(α) - cos^2(α)/sin^2(α)
= (sin^4(α)-cos^4(α)) / (cos^2(α) * sin^2(α))
= (sin^2(α) + cos2^(α))*(sin^2(α) - cos^2(α)) / (cos^2(α) * sin^2(α))
= (sin^2(α) - cos^2(α)) / (cos^2(α) * sin^2(α))
右边=-2*2 sin(2α)cos(2α) / sin(2α)sin^2(2α)
= -4cos(2α) / sin^2(2α)
=4(sin^2(α) - cos^2(α)) / 4(cos^2(α) * sin^2(α))
= (sin^2(α) - cos^2(α)) / (cos^2(α) * sin^2(α)) =左边