Question

If the identity $\cos^4\theta = a\cos4\theta + b\cos2\theta + c$ holds for some $a, b, c \in \mathbb{Q}$, then $(a, b, c) =$ ?

• A. $\left(\dfrac{1}{8}, \dfrac{3}{8}, \dfrac{1}{2}\right)$
• B. $\left(\dfrac{1}{8}, \dfrac{1}{2}, \dfrac{3}{8}\right)$
• C. $\left(\dfrac{1}{8}, \dfrac{1}{2}, \dfrac{1}{8}\right)$
• D. $\left(\dfrac{1}{8}, \dfrac{3}{8}, \dfrac{1}{8}\right)$

Hint:

Remember standard trigonometric identities for power of cosine functions.

Answer 1

The Correct Option is B

Use power-reduction identities: \[ \cos^4\theta = \dfrac{3}{8} + \dfrac{1}{2}\cos2\theta + \dfrac{1}{8}\cos4\theta \] Hence, comparing: \[ a = \dfrac{1}{8},\ b = \dfrac{1}{2},\ c = \dfrac{3}{8} \]