Answer
a) $C_1$ be any curve from $(0,0)$ to $(\pi,0)$
b) $C_2$ be any curve from $(0,0)$ to $(\dfrac{\pi}{2},0)$
(Other answers are also possible.)
Work Step by Step
a) Let us consider that $C_1$ be any curve from $(a,b)$ to $(c,d)$.
This implies that $\int_{C_1}F \cdot dr=\int_{C_1} \nabla F \cdot dr=f(c,d)-f(a,b)$
or, $\int_{C_1}F \cdot dr=\int_{C_1} \nabla F \cdot dr=f(r,s)-f(p,q)=\sin (r-2s)-\sin (p-2q)=0$
Thus, we have $a=0,b=0,c=\pi,d=0$and $C_1$ be any curve from $(0,0)$ to $(\pi,0)$
b) Let us consider that $C_2$ be any curve from $(a,b)$ to $(c,d)$.
This implies that $\int_{C_1}F \cdot dr=\int_{C_1} \nabla F \cdot dr=f(c,d)-f(a,b)$
or, $\int_{C_1}F \cdot dr=\int_{C_1} \nabla F \cdot dr=f(r,s)-f(p,q)=\sin (r-2s)-\sin (p-2q)=1$
and $a=0,b=0,c=\pi/2,d=0$
Thus, we have: $C_2$ can be any curve from $(0,0)$ to $(\dfrac{\pi}{2},0)$
