The magnitude of displacement of a particle moving in a circle of radius 'a' with constant angular speed 'w' varies with time 't' as:

R = acos@ (i) + asin@ (j)       ................1        

R is the position vector of particle at any time performing circular motion...

@ is angular displacement

w = @/t

@ = wt         ..............2

from 1 & 2

R_t = acoswt (i) + asinwt (j)          .............3

at t = 0 ,

R_o = a (i)             ..............4

displacement = R_t - R_o = a(coswt-1) (i) + a(sinwt) (j)

magnitude =[ (a²(coswt-1)² + a²(sinwt)²]^1/2

                =  [2a² - 2a²coswt]^1/2

                = a[2(1-coswt)]^1/2

now , coswt = (1-2cos²(wt/2)) , after putting this in above eq

  magnitude = 2a[cos²(wt/2)]^1/2

                  =2acos(wt/2)

this is the required result ,

approve if u like my ans

Approved

in the above answer it sayscoswt=1-cos^2(wt/2)but coswt = 1-sin^2(wt/2)=cos^2 (wt/2)-1also 2^(1/2) is not considered the final magnitude will be: (2)^(1/2)asin (wt/2)