Q5:

Lemma:

Prove of lemma:

(Induction) Cases n = 2, 3: Obvious by calculation. Suppose cases n = k – 2, k – 1 are true. When n = k,

Thus the lemma is true. Thus

Thus .

(It doesn’t need to be written in “expanded form”, does it? )

Q8:

Suppose x = a + bi, y = b + ci, z = c + ai. Hence: a = Re(x) b = Re(y) c = Re(z) And the original statement becomes:

Assume . Then:

.

Notice that since a,b,c > 0, we have .

Moreover, the cosine values do not change if we scale the whole system, so let’s assume |x|, |y|, |z|