Question

A true statement among the following identities is

• A. $\sin 5\theta = 16 \cos^4\theta \sin\theta - 12 \cos^2\theta \sin\theta + \sin\theta$
• B. $\sin 5\theta = 16 \cos^4\theta - 12 \cos^2\theta + 1$
• C. $\sin 5\theta = 16 \cos^4\theta \sin\theta + 12 \cos^2\theta \sin\theta - \sin\theta$
• D. $\sin 5\theta = 16 \cos^4\theta \sin\theta + \sin\theta$

Hint:

Memorize multiple angle identities or derive them using De Moivre’s theorem when needed.

Answer 1

The Correct Option is A

The identity for $\sin 5\theta$ in terms of $\sin\theta$ and $\cos\theta$ is:
$\sin 5\theta = 5\sin\theta - 20\sin^3\theta + 16\sin^5\theta$
Converting powers of $\sin\theta$ using $\sin^2\theta = 1 - \cos^2\theta$, we can expand and transform into the form:
$\sin 5\theta = \sin\theta(16 \cos^4\theta - 12 \cos^2\theta + 1)$
Which matches Option (1).