prove that cosA/1-sinA=2cosecA

cosA/(1-sinA) * (1+sinA)/(1+sinA)

= cosA(1+sinA)/(1-sin^2A)
= cosA(1+sinA)/cos^2A
= (1+sinA)/cosA
= secA + tanA

You can easily see that your original equation is false. Just use A = pi/4