Question

The value of $\frac{(\cos 2 \theta-i \sin 2 \theta)^4(\cos 4 \theta+i \sin 4 \theta)^{-5}}{(\cos 3 \theta+i \sin 3 \theta)^{-2}(\cos 3 \theta-i \sin 3 \theta)^{-9}}$ is

• A. $\cos 49 \theta-i \sin 49 \theta$
• B. $\cos 23 \theta-i \sin 23 \theta$
• C. $\cos 49 \theta+i \sin 49 \theta$
• D. $\cos 21 \theta+i \sin 21 \theta$

Answer 1

Answer: (A)

$\frac{(\cos 2 \theta-i \sin 2 \theta)^4(\cos 4 \theta+i \sin 4 \theta)^{-5}}{(\cos 3 \theta+i \sin 3 \theta)^{-2}(\cos 3 \theta-i \sin 3 \theta)^{-9}}$
$=e^{i-8 \theta-20 \theta+8 \theta-27 \theta) i}$
$=e^{(-49 \theta)i}=\cos 49 \theta-i \sin 49 \theta$