Since the problem says "show that the arc length... is..." we know what the answer should be at the end. It's always easier to get somewhere if we know where we are going. Let's deal with this like a parametric function. The parametric equations are
x(t) = f(t)cos t
y(t) = f(t)sin t
and the corresponding derivatives are
Use FOIL to square the derivatives. It's not as bad as it seems, because the blue terms cancel.
Now group the terms with (f'(t))2 together and the terms with (f(t))2 together.
Since sin2t + cos2t = 1, this simplifies to
the arc length of the curve for α ≤ t ≤ β is
This is equivalent to what we were supposed to get.