Question

A true statement among the following identities is:

• A. \( \cos 5\theta = 16 \cos^5\theta - 20 \cos^3\theta - 5 \cos\theta \)
• B. \( \cos 5\theta = 20 \cos^3\theta = 16 \cos^5\theta + 5 \cos\theta \)
• C. \( \cos 5\theta = 16 \cos^5\theta + 20 \cos^3\theta - 5 \cos\theta \)
• D. \( \cos 5\theta = 16 \cos^5\theta - 20 \cos^3\theta + 5 \cos\theta \)

Hint:

Remember key trigonometric identities for multiple angles like \( \cos 3\theta \), \( \cos 5\theta \), etc., especially in powers of cosine, as they are frequently used in expansions and transformations.

Answer 1

The Correct Option is D

We use the standard trigonometric identity for \( \cos 5\theta \) in terms of \( \cos\theta \): \[ \cos 5\theta = 16 \cos^5\theta - 20 \cos^3\theta + 5 \cos\theta \] This identity can be derived using De Moivre's Theorem or multiple angle formulas recursively. Thus, among all the given options, only Option (4) is correct.