Step 1: Note the setup.
Both $x$ and $y$ are given through a third variable $t$: $x=a(t+\sin t)$ and $y=a(1-\cos t)$. This is parametric form.
Step 2: Recall the parametric rule.
When $x$ and $y$ depend on $t$, \[ \frac{dy}{dx}=\frac{\,dy/dt\,}{\,dx/dt\,}. \] So we differentiate each with respect to $t$ and divide.
Step 3: Differentiate $x$.
\[ \frac{dx}{dt}=a(1+\cos t). \]
Step 4: Differentiate $y$.
\[ \frac{dy}{dt}=a(0+\sin t)=a\sin t. \]
Step 5: Divide to get $\frac{dy}{dx}$.
\[ \frac{dy}{dx}=\frac{a\sin t}{a(1+\cos t)}=\frac{\sin t}{1+\cos t}. \]
Step 6: Use half-angle identities.
Write $\sin t=2\sin\frac{t}{2}\cos\frac{t}{2}$ and $1+\cos t=2\cos^2\frac{t}{2}$: \[ \frac{dy}{dx}=\frac{2\sin\frac{t}{2}\cos\frac{t}{2}}{2\cos^2\frac{t}{2}}. \]
Step 7: Cancel and finish.
Cancel $2$ and one $\cos\frac{t}{2}$: \[ \frac{dy}{dx}=\frac{\sin\frac{t}{2}}{\cos\frac{t}{2}}=\tan\frac{t}{2}. \] This is option (1).
\[ \boxed{\tan\dfrac{t}{2}} \]