Let-S-0-pi-denote-the-set-of-values-of-x-satisfying-the-equation-8-1-cos-x-cos-2-x-cos-3-x-to-4-3-then-S-

Question Number 27203 by julli deswal last updated on 03/Jan/18

Let S \subset (0, π) denote the set of values of x

satisfying the equation

8^{1 + ∣ \cos x ∣ + \cos^{2} x + ∣ \cos^{3} x ∣ + \dots to \infty} = 4^{3}

then S =

Answered by Tinkutara last updated on 03/Jan/18

8^{\frac{1}{1 - ∣ \cos x ∣}} = 8^{2}

1 - ∣ \cos x ∣= \frac{1}{2}

\cos x = \pm \frac{1}{2}

S = {\frac{π}{3}, \frac{2 π}{3}}

Answered by mrW1 last updated on 03/Jan/18

4^{3} = 8^{2}

l e t t =∣ \cos x ∣

1 + t + t^{2} + t^{3} + \dots . = \frac{1}{1 - t} = 2

\Rightarrow t = \frac{1}{2}

\Rightarrow \cos x = \pm \frac{1}{2}

\Rightarrow x = \frac{π}{3} o r \frac{2 π}{3}

\Rightarrow S = {\frac{π}{3}, \frac{2 π}{3}}

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