**(a)** $$LHS=\frac{\sin^2A+(1+\cos A)^2}{\sin A(1+\cos A)}=\frac{\sin^2A+1+2\cos A+\cos^2A}{\sin A(1+\cos A)}=\frac{2+2\cos A}{\sin A(1+\cos A)}=\frac{2}{\sin A}=2\csc A=RHS$$

**(b)** $\sin(A+2B)=\frac{\sqrt3}{2} \Rightarrow A+2B=60°$. $\cos(A+4B)=0 \Rightarrow A+4B=90°$. Subtracting: $2B=30° \Rightarrow B=15°$, then $A=60-30=30°$. (Check: $A=30°>B=15°$ ✓)