Prove the following identities:`(1+cottheta-cosectheta)(1+tantheta+sectheta)=2`

Prove the following identities: (sintheta+sectheta)^2+(costheta+cosectheta)^2=(1+cosecthetasectheta)^2

Prove the following identities: (frac(cottheta)(cosectheta-1))=(frac(cosectheta+1)(cottheta))

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

Prove the following: (frac(1+cottheta+cosectheta)(1-cottheta-cosectheta))=(frac(cosectheta+cottheta-1)(cottheta-cosectheta+1))

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

Prove the following identities: 1+3cosec^2thetacot^2theta+cot^6theta=cosec^6theta

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

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

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

Prove the following: cottheta(1-cos2theta)=sin2theta