Question

If \( z = \cos\theta + i \sin\theta \), then the value of \( z^{100} + \frac{1}{z^{100}} \) is

• A. \( 2 \cos 100\theta \)
• B. \( 2^{100} \cos\theta \)
• C. \( 2i \sin 100\theta \)
• D. \( 2^{100} \sin\theta \)

Hint:

For complex numbers in polar form \( z = e^{i\theta} \), powers of \( z \) result in a rotation by multiples of \( \theta \). Use the Euler's formula to simplify such expressions.

Answer 1

The Correct Option is A

Step 1: Use Euler's formula.
We are given that \( z = \cos\theta + i \sin\theta \), which is the standard form of Euler’s formula. This means that \( z = e^{i\theta} \).

Step 2: Simplify the expression.
To find \( z^{100} + \frac{1}{z^{100}} \), we use the following property of complex exponentials: \[ z^{100} = e^{i100\theta} \quad \text{and} \quad \frac{1}{z^{100}} = e^{-i100\theta}. \] Thus, we have: \[ z^{100} + \frac{1}{z^{100}} = e^{i100\theta} + e^{-i100\theta} = 2 \cos(100\theta). \]
Step 3: Conclusion.
Thus, the value of \( z^{100} + \frac{1}{z^{100}} \) is \( 2 \cos 100\theta \), which corresponds to option (A).