First, let us get the coordinates for each point.
For point #f(pi/4)#, we have
#x_f = cos(pi/4)/2#
#y_f = sin^2(pi/4)#
As we know, #cos(pi/4) = sin(pi/4) = 1/sqrt(2)#
#x_f = 1/(2sqrt(2)) = sqrt(2)/4#
#y_f = (1/sqrt(2))^2 = 1/2#
For point #f(pi)# we have
#x_f = cos(pi)/2#
#y_f = sin^2(pi)#
We know that #cos(pi) = -1# and that #sin(pi) = 0#, so
#x_f = -1/2#
#y_f = 0#
The distance between #(sqrt(2)/4;1/2)# and #(-1/2;0)# is given by
#d^2 = (sqrt(2)/4 +1/2)^2 + (1/2)^2#
#d^2 = 1/16(sqrt(2)+2)^2 +1/4#
#d^2 = 1/16(2+4sqrt(2)+4) +1/4#
#d^2 = 1/16(6+4sqrt(2)) +1/4#
#d^2 = (3+2sqrt(2))/8 +1/4#
#d^2 = (3+2sqrt(2)+2)/8#
#d^2 = (5+2sqrt(2))/8#
#d = sqrt((5+2sqrt(2))/8)#
