How do I use the properties of inverses?

Question

Use properties of Inverses to verify the identity (assume all indicated operations are defined): A(A^-1 + B^-1)B(A+B)^-1= In

Answer 1

A(A^-1 + B^-1)B(A + B)^-1

= (AA^-1 + AB^-1)B(A + B)^-1 (Left-Distributivity)

= (I_n + AB^-1)B(A + B)^-1 (Inverses)

= (I_nB + AB^-1B)(A + B)^-1 (Right-Distributivity)

= (I_nB + AI_n)(A + B)^-1 (Inverses)

= (B + A)(A + B)^-1 (Identity)

= (A + B)(A + B)^-1 (Commutativity)

= I_n (Inverses)