If $0 \leq x < 2 π$ , then the number of real values of $x,$ which satisfy the equation $cos x + cos 2 x + cos 3 x + cos 4 x = 0$ :

3

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5

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7

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9

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Solution

The correct option is D $7$
$c o s x + c o s 4 x + c o s 2 x + c o s 3 x = 0$

$\Rightarrow 2 c o s (\frac{5 x}{2}) c o s (\frac{3 x}{2}) + 2 c o s (\frac{5 x}{2}) c o s (\frac{x}{2}) = 0$

$\Rightarrow 2 c o s (\frac{5 x}{2}) 2 c o s x c o s (\frac{x}{2}) = 0$

$c o s x = 0 \Rightarrow x = \frac{π}{2}, \frac{3 π}{2}$

$c o s \frac{x}{2} = 0 \Rightarrow x = π$

$c o s \frac{5 x}{2} = 0 \Rightarrow x = \frac{π}{5}, \frac{3 π}{5}, \frac{7 π}{5}, \frac{9 π}{5}$

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