Answer
The cyclist's required power output is 610 W.
Work Step by Step
If the cyclist coasts down the hill at a steady speed, then the force $F_R$ which opposes the motion must be equal in magnitude to $mg~sin(\theta)$.
This means that when the cyclist climbs the hill at a steady speed, the cyclist's force $F_p$ will be equal to the sum of opposing forces.:
$F_p = mg~sin(\theta) + F_R = 2mg~sin(\theta)$
We can use this force to find the cyclist's power.
$P = F_p\cdot v$
$P = 2mg~sin(\theta) \cdot v$
$P = (2)(75~kg)(9.80~m/s^2)~sin(6.0^{\circ}) \cdot (4.0~m/s)$
$P = 610~W$
Therefore, the cyclist's required power output is 610 W.