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

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

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

Prove the following: (frac(sin2x+sin2y)(sin2x-sin2y))=(frac(tan(x+y))(tan(x-y)))

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

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

Prove the following: (frac(sintheta)(1+costheta))=tan(theta/2)

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

Prove the following: frac(tan(theta/2)+cot(theta/2))(cot(theta/2)-tan(theta/2))=sectheta

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

Prove the following: (cosx+cosy)^2+(sinx-siny)^2=4cos^2(frac(x+y)(2))