Question

求证:tan^2x+cot^2x=2(3+cos4x)/(1-cos4x)

Answer 1

tan^2x+cot^2x=[(sinx)^4+(cosx)^4]/(sinxcosx)^2

={[(sinx)^2+(cosx)^2]^2-2(sinxcosx)^2}/(sinxcosx)^2

=[1-2(sinxcosx)^2]/(sinxcosx)^2

=4[1-1/2(sin2x)^2]/(sin2x)^2

=2[4-2(sin2x)^2]/(1-cos4x)

=2(3+cos4x)/(1-cos4x)

得证