Answer
$1$
Work Step by Step
Recall:
(1) $\cos{(90^\circ-\theta)}=\sin{\theta}$
(2) $\sin{(90^\circ-\theta)}=\cos{\theta}$.
(3) $\cos^2{\theta}+\sin^2{\theta}=1$
Thus,
$\cos{35^\circ}\sin{55^\circ}+\sin{35^\circ}\cos{55^\circ}\\
=\cos{35^\circ}\cos{(90^\circ-55^\circ)}+\sin{35^\circ}\sin{(90^\circ-55^\circ)}\\
=\cos{35^\circ}\cos{35^\circ}+\sin{35^\circ}\sin{35^\circ}\\
=\cos^2{35^\circ}+\sin^2{35^\circ}\\
=1$