Prove that `(cot^(2)theta)/(cosec theta+1)=cosec theta-1`

1+(cot^(2)theta)/(1+csc theta)=csc theta

Prove that: ("cosec"theta)/("cosec"theta-1)+("cosec"theta)/("cosec"theta+1)=2sec^(2)theta

Prove that (cot theta + "cosec" theta -1 )/(cot theta - "cosec" theta +1)=(1+ cos theta)/(sin theta).

prove that (1-cos theta)/(1+cos theta)=(csc theta-cot theta)^(2)

Prove that (cot theta+csc theta-1)/(cot theta-csc theta)=(cot theta+csc theta)(cot theta+csc theta-1)

Prove that cot^4 theta+ cot^2 theta= cosec^4 theta-cosec^2 theta

Prove that : (sin theta)/(1 - costheta) = cosec theta + cot theta

Prove that (sin theta)/(sec theta+tan theta-1)+cos theta/(cosec theta+cot theta-1)=1

Prove that (cos theta-sin theta+1)/(cos theta+sin theta-1)=csc theta+cot theta

Prove that: cosec theta +cosec (theta/2)+cosec (theta/2^2)+cosec(theta/2^3)+… to n terms = cot(theta/2^n) -cot theta