Prove the identity tanA/secA+1 + cotA/cosecA+1=cosecA+secA -secAtanA

Answer 1

only true if you meant:
tanA/(secA+1) + cotA/(cosecA+1)=cosecA+secA -secAtanA

LS = (sinA/cosA)(cosA) + 1 + (cosA/sinA)(sinA) + 1
= sinA + cosA + 2

RS = 1/sinA +1/cosA + (1/cosA)(sinA/cos)
= 1/sinA +1/cosA + (sinA/cos^2 A)
= (cos^2 A + sinAcosA + sin^2 A)/(sinAcos^2 A)
= (1 + sinAcosA)/(sinAcos^2 A)
not going anywhere ...

should have checked for some given angle in the original.
Just tried it with 20°, LS≠RS

you must have a typo, did you meant

tanA/(secA+1) + cotA/(cosecA+1) = cosecA+secA -secAtanA
or something like that ?