Prove the following identities:`(1+tan^2A)+(1+frac(1)(tan^2A))=frac(1)(Sin^2A-Sin^4A)`

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

Prove the following identities: (frac(1)(sectheta+tantheta))-(frac(1)(costheta))=(frac(1)(costheta))-(frac(1)(sectheta-tantheta))

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

Prove the following identities: (frac(sintheta)(1+costheta))+(frac(1+costheta)(sintheta))=2cosectheta

Prove the following: (frac(sec2A-1)(sec4A-1))=frac(tanA)(tan4A)

Prove that : (1+ (1)/(tan^2 A)) (1 +(1)/(cot^2 A)) = (1)/(sin^2 A - sin^4 A)

Prove the following: 2tan^-1(frac{1}{3}) = tan^-1(frac{3}{4})

Prove the following: (frac(tan5A-tan3A)(tan5A+tan3A))=(frac(sin2A)(sin8A))

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

Prove that (1+(1)/(tan^2 A)) (1+(1)/(cot^2A)) = (1)/(sin^2 A- sin^4 A)