Question:
A true statement among the following identities is:
A true statement among the following identities is:
Show Hint
Important trigonometric identities like
\[
\cos 3\theta
\quad \text{and} \quad
\cos 5\theta
\]
are frequently used in algebraic manipulations and equation solving.
Updated On: Jun 24, 2026
- \(\cos 5\theta = 16\cos^5\theta - 20\cos^3\theta - 5\cos\theta\)
- \(\cos 5\theta = 20\cos^3\theta - 16\cos^5\theta + 5\cos\theta\)
- \(\cos 5\theta = 16\cos^5\theta + 20\cos^3\theta - 5\cos\theta\)
- \(\cos 5\theta = 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta\)
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Recall the standard multiple-angle identity.
The standard identity for \(\cos 5\theta\) is \[ \cos 5\theta = 16\cos^5\theta -20\cos^3\theta +5\cos\theta \] This identity is obtained using De Moivre's theorem or repeated angle transformations.
Step 2: Compare with the given options.
Now compare the standard identity with each option.
Option (4) exactly matches: \[ \cos 5\theta = 16\cos^5\theta -20\cos^3\theta +5\cos\theta \] Hence, option (4) is correct.
Step 3: Final conclusion.
Therefore, the true identity is \[ \boxed{\cos 5\theta = 16\cos^5\theta -20\cos^3\theta +5\cos\theta} \]
The standard identity for \(\cos 5\theta\) is \[ \cos 5\theta = 16\cos^5\theta -20\cos^3\theta +5\cos\theta \] This identity is obtained using De Moivre's theorem or repeated angle transformations.
Step 2: Compare with the given options.
Now compare the standard identity with each option.
Option (4) exactly matches: \[ \cos 5\theta = 16\cos^5\theta -20\cos^3\theta +5\cos\theta \] Hence, option (4) is correct.
Step 3: Final conclusion.
Therefore, the true identity is \[ \boxed{\cos 5\theta = 16\cos^5\theta -20\cos^3\theta +5\cos\theta} \]
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