Use De Moivre's Theorem to prove this:

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In summary, by using De Moivre's Theorem and converting tanθ into \frac{sin\theta}{cos\theta}, it can be shown that (1+itanθ)^n + (1-itanθ)^n = 2cosnθ/cosnθ, where cosθ ≠ 0. This is a common question in the New South Wales HSC, and the solution involves converting the expression into modulus argument form and simplifying from there. Good luck with your exams!
  • #1
aanandpatel
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Use De Moivre's Theorem to show that for any n greater that equal to 1

(1+itanθ)n + (1-itanθ)n =2cosnθ/cosnθ

where cosθ ≠ 0


I tried to approach this by converting into modulus argument form but wasn't really sure if that was correct. It's a common New South Wales HSC question but I couldn't find a solution anywhere. Help would be greatly appreciated :)
 
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  • #2
The first step would be to convert tanθ into [itex]\frac{sin\theta}{cos\theta}[/itex] and work from there.
 
  • #3
Thanks a bunch - that helped a lot. Converted it into:
[(secθ)(cosθ+isinθ)]^n + [(secθ)(cosθ-isinθ)]^n and it was easy from there.

Cheers!
 
  • #4
No problem. Good luck with the HSC, I just finished mine :P
 

What is De Moivre's Theorem?

De Moivre's Theorem is a mathematical formula that allows us to raise a complex number to a power. It states that for any complex number z and any positive integer n, (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ).

How do you use De Moivre's Theorem?

To use De Moivre's Theorem, we need to first convert the complex number into polar form, with the angle θ and the modulus r. Then, we raise the modulus to the power of n and multiply the angle by n. Finally, we convert the result back to rectangular form.

What are the applications of De Moivre's Theorem?

De Moivre's Theorem has many applications in mathematics and engineering, particularly in solving problems involving complex numbers, trigonometry, and geometry. It is also used in signal processing, electrical engineering, and quantum mechanics.

How does De Moivre's Theorem relate to Euler's Formula?

Euler's Formula, e^(ix) = cos(x) + i sin(x), is a special case of De Moivre's Theorem when n = 1. De Moivre's Theorem can be derived from Euler's Formula by using the binomial theorem and expanding the complex number (cos θ + i sin θ)^n.

Can De Moivre's Theorem be used to prove other theorems?

Yes, De Moivre's Theorem can be used to prove other theorems in mathematics, such as the trigonometric addition formulas. It can also be used to prove properties of complex numbers, such as the nth root theorem and the fundamental theorem of algebra.

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