The number of values of x lying in the internal (−π,π) which satisfy the equation 81+|cos⁡x|+cos2⁡x+cos3⁡x+…∞=43,is

# The number of values of x lying in the internal $\left(-\mathrm{\pi },\mathrm{\pi }\right)$ which satisfy the equation ${8}^{\left(1+|\mathrm{cos}\mathrm{x}|+{\mathrm{cos}}^{2}\mathrm{x}+\left|{\mathrm{cos}}^{3}\mathrm{x}\right|+\dots \mathrm{\infty }\right)}={4}^{3},$is

1. A

3

2. B

4

3. C

5

4. D

6

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### Solution:

Now,$1+|\mathrm{cos}\mathrm{x}|+{\mathrm{cos}}^{2}\mathrm{x}+\left|{\mathrm{cos}}^{3}\mathrm{x}\right|+\dots \mathrm{\infty }=\frac{1}{1-|\mathrm{cos}\mathrm{x}|}$

$\begin{array}{ll}⇒& \frac{3}{1-|\mathrm{cos}\mathrm{x}|}={2}^{6}⇒1=2-2|\mathrm{cos}\mathrm{x}|\\ ⇒& |\mathrm{cos}\mathrm{x}|=\frac{1}{2}\\ ⇒& \mathrm{cos}\mathrm{x}=±\frac{1}{2}\\ ⇒& \mathrm{x}=\frac{\mathrm{\pi }}{3},-\frac{\mathrm{\pi }}{3},\frac{2\mathrm{\pi }}{3},-\frac{2\mathrm{\pi }}{3}\end{array}$

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