求证：tan2x+1tan2x=2(3+cos4x)1−cos4x．

Question

求证：tan²x+(1)/(tan2x)=(2(3+cos4x))/(1−cos4x)．

Answer 1

证明：左边=(sin2x)/(cos2x)+(cos2x)/(sin2x)
=(sin4x+cos4x)/(sin2xcos2x)
=((sin2x+cos2x)2−2sin2xcos2x)/((1)/(4)sin22x)
=(8−4sin22x)/(1−cos4x)=(4+4cos22x)/(1−cos4x)
=(4+2(1+cos4x))/(1−cos4x)=(2(3+cos4x))/(1−cos4x)
=右边．
∴tan²x+(1)/(tan2x)=(2(3+cos4x))/(1−cos4x)．