This is not identity.
If we have 1/(secA+tanA)=(1-secA)/cosA then it is equal
If we put A=60 degrees then cosA=1/2
In left part of identity we'll get 1/(sqrt(3)+1) and for
the right part -2. They are not equal.
If we omit parenthesis
1/secA+tanA=1-secA/cosA then it's equivalent to (cos^2(A)+sinA)/cosA=
Similar way we'll get again not equal parts.
it is error in given statement.