If $$cos$$⁡$$4x=a0+a1cos2$$⁡$$x+a2cos4$$⁡x s true for all values of x∈R then the value of $$5a0+a1+a2$$ is

Multiple and sub- multiple Angles

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Question

If $\cos 4 x = a_{0} + a_{1} \cos^{2} x + a_{2} \cos^{4} x$ s true for all values of $x \in R$ then the value of $5 a_{0} + a_{1} + a_{2}$ is

Moderate

Solution

$\begin{array}{l} \cos 4 x = 2 \cos^{2} 2 x - 1 \\ = 2 {(2 \cos^{2} x - 1)}^{2} - 1 \\ = 2 (4 \cos^{4} x + 1 - 4 \cos^{2} x) - 1 \\ = 8 \cos^{4} x - 8 \cos^{2} x + 1 \\ ∴ a_{0} = 1, a_{1} = - 8, a_{2} = 8 \\ ∴ 5 a_{0} + a_{1} + a_{2} = 5 \end{array}$

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