For the first problem you start with equations that describe the fact that the spherical light wave illuminates the surface of sphere ct meters in radius. Thus for every point on the sphere we have xsqrd+ysqrd +z sqrd = r sqrd and since the light wave moves outward ct meters per second the radius is ct. One equation describes the events seen in K the other in k. I asked you to demonstrate that the events seen in one frame are consistent with the measurements of those events in the other frame. So if someone in K sees point x y z get hit by the light wave at time t, someone in k will see the same event and describe it as happening at t', x', y' and z'. In other words show that if point t,x,y,z, complies with the equation in K, then its transform in k complies with the equation in k. One way to do this is to take the k equation and write the four quantities as the transformed quantities from K, (i.e. replace t' with the correct function of phi, gamma,t,v,x, , x' with the correct function of phi, gamma, t,v,x etc. using equations (6). Then simplify this equation to see if gets you back to the K equation.
Hints
1) divide through by phi sqrd since it appears in every term.
2) gather all the t terms on one side and all the x terms on the other.
3) factor out csqrd t sqrd from one parentheses on one side
4) factor out xsqrd
5) remember what gamma sqrd equals
