Question:
If the identity $\cos^4\theta = a\cos4\theta + b\cos2\theta + c$ holds for some $a, b, c \in \mathbb{Q}$, then $(a, b, c) =$ ?
If the identity $\cos^4\theta = a\cos4\theta + b\cos2\theta + c$ holds for some $a, b, c \in \mathbb{Q}$, then $(a, b, c) =$ ?
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Remember standard trigonometric identities for power of cosine functions.
Updated On: May 18, 2025
- $\left(\dfrac{1}{8}, \dfrac{3}{8}, \dfrac{1}{2}\right)$
- $\left(\dfrac{1}{8}, \dfrac{1}{2}, \dfrac{3}{8}\right)$
- $\left(\dfrac{1}{8}, \dfrac{1}{2}, \dfrac{1}{8}\right)$
- $\left(\dfrac{1}{8}, \dfrac{3}{8}, \dfrac{1}{8}\right)$
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The Correct Option is B
Solution and Explanation
Use power-reduction identities:
\[
\cos^4\theta = \dfrac{3}{8} + \dfrac{1}{2}\cos2\theta + \dfrac{1}{8}\cos4\theta
\]
Hence, comparing:
\[
a = \dfrac{1}{8},\ b = \dfrac{1}{2},\ c = \dfrac{3}{8}
\]
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