Answer
$$1$$
Work Step by Step
Recall the co-function identity:
$ \sin (\theta)=\cos(90^{\circ}-\theta)$ and $ \cos (\theta)=\sin (90^{\circ}-\theta)$
We know that
$\sin^2 \theta+\cos^2 \theta=1$
Therefore, $$\cos \ 35^{\circ} \sin 55^{\circ}+\sin 35^{\circ} \ \cos \ 55^{\circ} =\cos 35^{\circ} \ \cos (90^{\circ}-55^{\circ}) +\sin 35^{\circ} \sin (90^{\circ}-55^{\circ}) \\=\cos \ 35^{\circ} \cos 35^{\circ}+\sin 35^{\circ} \ \sin \ 35^{\circ} \\=\cos^2 35^{\circ}+\sin^2 35^{\circ} \\=1$$
