$\begin{align*}
&(A) \\ \\
&R = {\color{red}{(a^*b)^*}}a^* \\
&R = {\color{red}{R_1}}a^* \\
\\
&\text{Now, } \\
&{\color{red}{R_1}} = {\color{red}{(a^*b)^*}} \\
&{\color{red}{R_1}} = \left [ b+ab+aab+aaab+aaaab+... \right ]^* = \epsilon +{\color{red}{(a+b)^*}}b \quad \dots (1)\\
\\
&\text{Again, } \\
&{\color{red}{R_1}} = {\color{red}{(a^*b)^*}} \\
&{\color{red}{R_1}} = \epsilon + a^*b(a^*b)^* \\
&{\color{red}{R_1}} = \epsilon + (a^*b)^*a^*b \qquad \left [ rr^* = r^*r \right ]\\
&{\color{red}{R_1}} = \epsilon + \left [ {\color{red}{(a^*b)^*a^*}} \right ]b \quad \dots (2)\\
\\
&\text{Compare} (1) \text{ and } (2) \Rightarrow {\color{red}{(a+b)^*}} = {\color{red}{(a^*b)^*a^*}}\\
\\
\hline \\
&(B) \\ \\
&R = {\color{red}{(b^*a)^*}}b^* \\
&R = {\color{red}{R_1}}b^* \\
\\
&\text{Now, } \\
&{\color{red}{R_1}} = {\color{red}{(b^*a)^*}} \\
&{\color{red}{R_1}} = \left [ a+ba+bba+bbba+bbbba+... \right ]^* = \epsilon +{\color{red}{(a+b)^*}}a \quad \dots (1)\\
\\
&\text{Again, } \\
&{\color{red}{R_1}} = {\color{red}{(b^*a)^*}} \\
&{\color{red}{R_1}} = \epsilon + b^*a(b^*a)^* \\
&{\color{red}{R_1}} = \epsilon + (b^*a)^*b^*a \qquad \left [ rr^* = r^*r \right ]\\
&{\color{red}{R_1}} = \epsilon + \left [ {\color{red}{(b^*a)^*b^*}} \right ]a \quad \dots (2)\\
\\
&\text{Compare} (1) \text{ and } (2) \Rightarrow {\color{red}{(a+b)^*}} = {\color{red}{(b^*a)^*b^*}}\\
\\
\hline \\
&(C) \\ \\
&(a^*b)^*a^* = a^*(ba^*)^* \qquad \text{because : }(pq)^*p = p(qp)^*\\
\\
\hline \\
&(D) \\ \\
&(b^*a)^*b^* = b^*(ab^*)^* \qquad \text{because : }(pq)^*p = p(qp)^*\\
\end{align*}$
