Question

A particle is moving around in a circle and its position is given in polar coordinates as x = Rcosθ, and y = Rsinθ, where R is the radius of the circle, and θ is in radians. From these equations derive the equation for centripetal acceleration.

Answer 1

Your motion has been shown in the figure. I have assumed that motion started from (R,0).
Although result will be same if you choose any other initial position.ω is the angular velocity.
So, θ =ωt in time t

position vector at any time,t is therefore given by
r=Rcosωt i +Rsinωt j
i and j are unit vectors along x and y axis respectively.

Differentiating twice , will gives us acceleration

a=-ω^2r
You can verify this yourself by differentiation.
Since there is no other acceleration ,this being uniform circular motion.This total acceleration is equal to centripetal acceleration.