Prove that`(frac(tan^3 x)(1+tan^2 x))+(frac(cot^3 x)(1+cot^2 x))=secxcosecx-2sinxcosx`

Prove the following identities: (frac(tan^3theta)(1+tan^2theta))+(frac(cot^3theta)(1+cot^2theta))=secthetacosectheta-2sinthetacostheta

Prove the following: (frac(sin3x)(cosx))+(frac(cos3x)(sinx))=2cot2x

Prove the following: (frac(1)(tan3A-tanA))-(frac(1)(cot3A-cotA))=cot2A

tan^(-1) (cot x) +cot^(-1)(tan x) =

Prove the following: (frac(cosx)(1+sinx))=(frac(cot(x/2)-1)(cot(x/2)+1))

Prove the following: (frac(1+tanx)(1-tanx))^2=(frac(tan(pi/4+x))(tan(pi/4-x)))

Prove the following: sinxtan(x/2)+2cosx=(frac(2)(1+tan^2(x/2)))

Prove the following: (frac(cosx+sinx)(cosx-sinx))-(frac(cosx-sinx)(cosx+sinx))=2tan2x

Prove the following: (frac(cot6A-tan2A)(cot6A+tan2A))=(cos8A)/(cos4A)

Prove the following: sqrt(frac(1+sin2x)(1-sin2x))=tan(pi/4+x)