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  4. The number of solutions of displaystyle∑r=14 cos r x=4 in the interval [0,2 π] is

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Q. The number of solutions of $\displaystyle\sum_{r=1}^{4} \cos r x=4$ in the interval $[0,2 \pi]$ is

KEAMKEAM 2019

Solution:

$\displaystyle\sum_{r=1}^{4} \cos r x=4$
$\cos x+\cos 2 x+\cos 3 x+\cos 4 x=4$
Which is possible only iff,
$\cos x=\cos 2 x=\cos 3 x=\cos 4 x=\cos 5 x=1$
and it satisfied by $x=0$ only
$\therefore $ Number of solutions is $1$ .