Question

guys can someone prove demoivre's theorem​

Answer 1

Answer:

De Moivre’s Theorem Proof

Apply Mathematical Induction to prove De Moivre’s Theorem.

We know, (cos x + i sin x)n = cos(nx) + i sin(nx) …(i)

Step 1: For n = 1, we have

(cos x + i sin x)1 = cos(1x) + i sin(1x) = cos(x) + i sin(x)

Which is true.

Step 2: Assume that formula is true for n = k.

(cos x + i sin x)k = cos(kx) + i sin(kx) ….(ii)

Step 3: Prove that result is true for n = k + 1.

(cos x + i sin x)k+1 = (cos x + i sin x)k (cos x + i sin x)

= (cos (kx) + i sin (kx)) (cos x + i sin x) [Using (i)]

= cos (kx) cos x − sin(kx) sinx + i (sin(kx) cosx + cos(kx) sinx)

= cos {(k+1)x} + i sin {(k+1)x}

=> (cos x + i sin x)k+1 = cos {(k+1)x} + i sin {(k+1)x}

Hence the result is proved.