a) You are given:

z=cos(theta)+i.sin(theta)

so:

|z|=sqrt[cos^2(theta)+sin^2(theta)]

but cos^2(theta)+sin^2(theta)=1, so |z|=1.

b) 1/z=1/[cos(theta)+i.sin(theta)]

now multiply top and bottom on the right by cos(theta)-i.sin(theta) to get:

1/z=(cos(theta)-i.sin(theta)/[(cos(theta)+i.sin(theta))(cos(theta)-i.sin(theta))]

.....=(cos(theta)-i.sin(theta)/[cos^2(theta)+sin^2(theta))]

.....=cos(theta)-i.sin(theta)

Now cos(theta)=cos(-theta), and sin(theta)=-sin(-theta), so putting these into the last equation above gives:

1/z=cos(-theta)+i.sin(-theta)

as required.

RonL