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(1)let us consider the case of a unit Sphere { as given in question} with the Reoper interval as :-ds^(2)=dtheta^(2)+sin^(2)theta dphi^(2)" (A) "Step 1:- Extract the lagrangian from the given equation of fuoper intorval i.e Equation (1):.quad L=theta^(2)+sin^(2)thetaphi^(2)Step 2:- lars this lagrangian through the Euler - Lagrange Equation(d)/(d lambda)((del L)/(del theta))=(del L)/(del theta)which gives theta-(sin theta cos theta)phi^(2)=0Similarly for the phi component results inphi+2(cot theta)theta phi=0-(3)Step 3:- Take the Geodesic equation and write it out for each of the components:Geadesic Gquation; x^(¨)i^(˙)+Gamma_(jk)^(i)x^(˙)^(-j)x^(k)=0 if x^(1)=theta and x^(2)=O/(2)then for i=1;theta^(¨)+Gamma_(theta theta)^(theta ... See the full answer