(a) Prove that (cos i sin )2 cos 2 i sin 2, where i C and i2 1 (b) Using induction, prove that for all n Z , (cos i sin )n cos n i sin n (This result is known as DeMoivre 's Theorem ) (c) Verify that 1 i 2(cos 45 i sin 45), and compute (l i)100 ...

a cos sin 2 cos 2 2isincos sin 2 cos 2 sin 2 t2sincos cos 2 a sin 2 b Sn cos sin n cos n i sin n S1 ...

(a) Prove that (cos + i sin )2 = cos 2 + i sin 2, where...

Question:

(a) Prove that (cos θ + i sin θ)2 = cos 2θ + i sin 2θ, where i ∈ C and i2 = -1.
(b) Using induction, prove that for all n ∈ Z+,
(cos θ + i sin θ)n = cos nθ + i sin nθ.
(This result is known as DeMoivre's Theorem.)
(c) Verify that 1 + i = √2(cos 45° + i sin 45°), and compute (l + i)100.