Trigonometric equations

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Question

Let $θ \in [0, 4 π]$ satisies the equation $(\cos θ + 1) (\cos θ + 2) (\cos θ + 3) = 24$ , then the sum of all the values of $θ$ is of the form $k π$ then K =

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Solution

$T h e m a x i m u m v a l u e o f \cos θ i s 1$
We have $L H S \leq 24, R H S = 24$
$∴ L H S = R H S = 24$
$\Rightarrow \cos θ = 1 \Rightarrow θ = 0, 2 π, 4 π$

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