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

f ( 0 ) = 1 8 , maximum value of f ( θ ) ∀ θ ∈ R is 1 8 , number of principle solutions of f ( θ ) = 0 is 8

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

f ( θ ) = ( cos − cos π 8 ) ( cos θ − cos 3 π 8 ) ( cos θ − 5 π 8 ) ( cos θ − 7 π 8 ) ⇒ f ( θ ) = ( cos 2 θ − cos 2 π 8 ) ( cos 2 − cos 2 3 π 8 ) Simplifying we get f ( θ ) = 1 8 cos 4 θ

maximum value of f ( θ ) ∀ θ ∈ R is 1 4

maximum value of f ( θ ) ∀ θ ∈ R is 1 8

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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
$f (θ) = \frac{1}{8} \cos 4 θ$