R - radius of the larger circle
r - radius of the smaller circle
Circumference of the larger circle =

Circumference of the smaller circle =

Part 2: During one complete rotation of the larger gear both gears pass the same distance

.
Therefore, the smaller gear makes

complete rotations.
Part 1: During one complete rotation of the smaller gear both gears pass the same distance

.
Therefore, the larger gear makes

of a complete rotation.
Since one complete rotation corresponds to 360 degrees, 1/4 of a complete rotation corresponds to 360/4=90 degrees.