The formulas means:

• two complex numbers are multiplied by multiplying their length and adding their angles
• a complex number is raised to a power by raising the length to the power and multiplying the angle

Multiplying vectors z and w

The proof is an exercise in trig identities. Recall that:

(1)       cos(x + y) = cos(x)cos(y)-sin(x)sin(y)
(2)       sin(x + y) = sin(x)cos(y)+cos(x)sin(y)

Now

z*w = r₁ cis(s) r₂ cis(t) =
     = r₁(cos(s)+ i sin(s)) r₂(cos(t) + i sin(t)) =
     = r₁ r₂ (cos(s)+ i sin(s))(cos(t) + i sin(t)) =
     = r₁ r₂ (cos(s) cos(t) - sin(s)sin(t)) + i(cos(s)sin(t) + sin(s)cos(t)) =
     = r₁ r₂ (cos(s+t) + i sin(s+t)) =
     = r₁ r₂ cis(s+t)

which proves the first assertion. As for the second, first note that

z² = z*z = r cis(t) r cis(t) = r² cis(2t)

by our previous result. Now you should be able to finish the proof by carefully writing down an induction proof yourself.