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The most general solution of
$2^{1 + | \cos x | + \cos^{2} x + |\cos^{3} x| + \dots = 4}$ is given by

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a

x = n π \pm \frac{π}{3}, n \in Z

b

z = 2 n π \pm \frac{π}{3} n \in Z

c

x = 2 n π \pm \frac{2 π}{3}, n \in Z

d

none of these

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detailed solution

Correct option is A

The given equation is not meaningful for

$| \cos x | = 1 . So, let | \cos x | \neq 1 .$

Now,

$2^{1 + | \cos x | + \cos^{2} x + | \cos x |^{3} + \dots + tos x} = 4$

$\begin{array}{l} \Rightarrow \frac{1}{2^{1 - | \cos x |}} = 2^{2} \\ \Rightarrow \frac{1}{1 - | \cos x |} = 2 \\ \Rightarrow 2 - 2 | \cos x | = 1 \\ \Rightarrow | \cos x | = \frac{1}{2} \\ \Rightarrow \cos x = \pm \frac{1}{2} \\ \Rightarrow \cos x = \cos \frac{π}{3}, \cos x = \cos \frac{2 π}{3} \\ \Rightarrow x = 2 n π \pm \frac{π}{3}, x = 2 n π \pm \frac{2 π}{3}, n \in Z \\ \Rightarrow x = 2 n π \pm \frac{π}{3}, x = (2 n \pm 1) π \pm \frac{π}{3} \\ \Rightarrow x = n π \pm \frac{π}{3}, n \in Z . \end{array}$

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