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 possible.)
Work Step by Step
a) Suppose $C_1$ be any curve from $(p,q)$ to $(r,s)$.
Then, we have $\int_{C_1}F \cdot dr=\int_{C_1} \nabla F \cdot dr=f(r,s)-f(p,q)$
This implies that $\int_{C_1}F \cdot dr=\int_{C_1} \nabla F \cdot dr=f(r,s)-f(p,q)=\sin (r-2s)-\sin (p-2q)=0$
Here, we have $p=0,q=0,r=\pi,s=0$
Thus, $C_1$ be any curve from $(0,0)$ to $(\pi,0)$
b) Suppose $C_2$ be any curve from $(p,q)$ to $(r,s)$.
Then, we have $\int_{C_1}F \cdot dr=\int_{C_1} \nabla F \cdot dr=f(r,s)-f(p,q)$
This implies that $\int_{C_1}F \cdot dr=\int_{C_1} \nabla F \cdot dr=f(r,s)-f(p,q)=\sin (r-2s)-\sin (p-2q)=1$
Here, we have $p=0,q=0,r=\pi/2,s=0$
Thus, $C_2$ be any curve from $(0,0)$ to $(\dfrac{\pi}{2},0)$
