Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$$\frac { \sin ( 3 \theta ) } { \sin \theta } = ?$$
A) $2 \sin \theta \cos \theta$
B) $\cos ( 2 \theta ) + \sin \theta \cos \theta$
C) $2 \cos ( 2 \theta )$
D) $4 \cos ^ { 2 } \theta - 1$

Question

Quizplus · Accepted Answer

The identity $\frac { \sin ( 3 \theta ) } { \sin \theta } = 4 \cos ^ { 2 } \theta - 1$ can be derived using the triple angle formula for sine, $\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)$, and dividing both sides by $\sin(\theta)$, simplifying to $3 - 4\sin^2(\theta)$. Using the Pythagorean identity, $\sin^2(\theta) + \cos^2(\theta) = 1$, we can replace $\sin^2(\theta)$ with $1 - \cos^2(\theta)$, leading to $3 - 4(1 - \cos^2(\theta)) = 4\cos^2(\theta) - 1$.

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