Question

If \( |Z|=2 \), \( Z_1 = \frac{Z}{2}e^{i\alpha} \) and \( \theta \) is the amp(Z), then \( \frac{Z_1^n - Z_1^{-n}}{Z_1^n + Z_1^{-n}} = \)

• A. \( 2^n i \tan(n\theta + n\alpha) \)
• B. \( i \tan(n\theta - n\alpha) \)
• C. \( i \tan(n\theta + n\alpha) \)
• D. \( \tan(n\theta + n\alpha) \)

Hint:

Recall that \( \frac{e^{ix} - e^{-ix}}{e^{ix} + e^{-ix}} = \frac{2i\sin x}{2\cos x} = i\tan x \). Recognizing this hyperbolic-like structure with complex exponents allows for instant simplification.

Answer 1

The Correct Option is C