Question

To prove -

(CosecA/CosecA-1)+(CosecA/CosecA+1)=2secA

Answer 1

Your questions has a slight mistake .

Actually
(cosec A/cosec A-1)+
(cosec A/cosec A+1)
=2sec²A
RHS is not 2 secA,it is sec²A

Divide throughout by cosec A

LHS becomes
(1/1-sinA)+(1/1+sinA)
=2/(1-sin²A)
=2/cos²A
=2sec²A
=RHS