Let f(θ)=(cosθ−cosπ8)(cosθ−cos3π8)(cosθ−cos5π8)(cosθ−cos7π8) then:

f(θ)=(cos−cosπ8)(cosθ−cos3π8) (cosθ−5π8)(cosθ−7π8) ⇒f(θ)=(cos2θ−cos2π8)(cos2−cos23π8)Simplifying we get f(θ)=18cos4θ

Let f(θ)=(cosθ−cosπ8)(cosθ−cos3π8)(cosθ−cos5π8)(cosθ−cos7π8) then:

f(θ)=(cos−cosπ8)(cosθ−cos3π8) (cosθ−5π8)(cosθ−7π8) ⇒f(θ)=(cos2θ−cos2π8)(cos2−cos23π8)Simplifying we get f(θ)=18cos4θ

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Let $f (θ) = (\cos θ - \cos \frac{π}{8}) (\cos θ - \cos \frac{3 π}{8}) (\cos θ - \cos \frac{5 π}{8}) (\cos θ - \cos \frac{7 π}{8})$ then:

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maximum value of $f (θ) \forall θ \in R$ is $\frac{1}{4}$

maximum value of $f (θ) \forall θ \in R$ is $\frac{1}{8}$

$f (0) = \frac{1}{8}$

number of principle solutions of $f (θ)$ = 0 is 8

answer is C, B, D.

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Detailed Solution

$f (θ) = (\cos - \cos \frac{π}{8}) (\cos θ - \cos \frac{3 π}{8})$
$(\cos θ - \frac{5 π}{8}) (\cos θ - \frac{7 π}{8})$
$\Rightarrow f (θ) = (\cos^{2} θ - \cos^{2} \frac{π}{8}) (\cos^{2} - \cos^{2} \frac{3 π}{8})$
Simplifying we get
$f (θ) = \frac{1}{8} \cos 4 θ$