Proof of De Moivre's Theorem (DP IB Analysis & Approaches (AA)): Revision Note

Proof of De Moivre's theorem

How do I prove de Moivre’s theorem?

• You only need to be able to prove de Moivre's theorem for , i.e. positive integer values of

  • in which case you can use proof by induction

• You need to prove that for

• STEP 1
  Prove it is true for

  • 

  • So de Moivre’s Theorem is true for

• STEP 2
  Assume it is true for

  • 

• STEP 3
  Show it is true for

  • 

  • According to the assumption this is equal to

    • 

  • Using laws of indices and multiplying out the brackets

    • 

  • Using and collecting the real and imaginary parts gives

    • 

  • Recognising that the real part is equivalent to and the imaginary part is equivalent to gives

    • 

  • So de Moivre’s Theorem is true for

• STEP 4
  Write a conclusion to complete the proof

  • The statement is true for , and if it is true for it is also true for

  • Therefore, by the principle of mathematical induction, the result is true for all positive integers,