A portion of a homework problem was given me to solve for practice. I have solved some but not all of the homework problem and I hope you all can help.

Here is the problem:

1. For each x R it is conventional to write cis(x) = cos(x) + i sin(x). Prove that cis(x+y) = cis(x) cis(y).

Let x, y R. We want to prove that cis(x+y) = cis(x) cis(y). Thus,

cis(x+y) = cos(x+y) + i sin(x+y)

= (cos(x) cos(y) - sin(x)sin(y)) + i(cos(x)sin(y) + sin(x)cos(y))

= cos(x) cos(y) - sin(x) sin(y) + i sin(x) cos(y) + i sin(y) cos(x)

= (cos(x)+ i sin(x))(cos(y) + i sin(y))

= cis(x) cis(y)

2. Let T designate the set {cis(x) : x R}, that is, the set of all the complex numbers lying on the unit circle, with the operation of multiplication. Use part 1 to prove that T is a group.

3. Use the FHT to conclude that T isomorphic R/<2>

4. Prove that g(x) = cis(2x) is a homomorphism from R onto T, with kernel Z

Let g: R -> T by g(x) = cis(2x).

g is subjective since ever element of T is of the from cis(a) = cis(2(a/2)) = g(a/2) for some a R. The kernel of g is the set of x R such that cis(2x) = 1. This equation only holds true if and only if 2x = 2k for some k Z. Divide by 2 then you get ker(g) = Z. Hence, g is a homomorphism with a kernel of Z

5. Conclude that T is isomorphic R/Z

By the FHT g: R -> T is a homomorphism and R is subjective to T. Since Z is the kernel of g then H is isomorphic to R/Z

I really have no clue how to do 2 and 3 so any help would be great on that. Also if you can verify that my other three are correct that would be great. Thanks for the help in advance