Question

(tan A + sec(A) - 1)/(tan A - SecA) = (1 + sin A)/(cos A)​

Answer 1

Solution:
LHS

= tan(a)+sec(a)-1/tan(a)-sec(a)

= (sin(a)/cos(a))+(1cos(a))-1/(sin(a)/cos(a))-(1/cos(a))

= sin(a)+1-cos(a)/cos(a)/sin(a)/cos(a)-1/cos(a)

= sin(a)+1-cos(a)/cos(a)/sin(a)-1/cos(a)

= sin(a)+1-cos(a)/ sin(a)-1

RHS

= 1+sin(a)/cos(a)

= 1/cos(a) + sin(a)/cos(a)

= sec(a)+tan(a)

The result is not proven.

Hope this will help.