Step 1: Know the goal. We must add four trigonometric terms that all involve the same angle $\theta$. The smart move is to change every term into a plain $\sin\theta$ using the standard angle rules, so they can cancel.
Step 2: Use the rule for $\cos(90^\circ+\theta)$. Adding $90^\circ$ turns cosine into a negative sine. \[ \cos(90^\circ+\theta)=-\sin\theta \]
Step 3: Use the rule for $\sin(180^\circ-\theta)$. In the second quadrant sine stays positive. \[ \sin(180^\circ-\theta)=\sin\theta \]
Step 4: Use the rule for $\sin(180^\circ+\theta)$. In the third quadrant sine becomes negative. \[ \sin(180^\circ+\theta)=-\sin\theta \]
Step 5: Substitute everything in. Replace each term in the original expression. \[ \sin\theta+(-\sin\theta)+\sin\theta+(-\sin\theta) \]
Step 6: Cancel and finish. The first and second cancel, and the third and fourth cancel, leaving nothing. \[ =0 \] So the whole sum is $0$, which is option 1. \[ \boxed{0} \]