Find x in (-π, π) where 8^(1+|cos x|+cos²x+...) = 4³

[anemone]

Hello MHB,

I am truly honored and delighted to have been asked to take over posting the POTW for Secondary School/High School Students. This week's problem is as follows:

Find all the values of $x$ lying in the interval $(-\pi,\,\pi)$ which satisfy the equation

$8^{1+|\cos x|+\cos^2 x+|\cos^3 x|+\cdots+\infty}=4^3$--------------------
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to the following members for their correct solutions::)

1. kaliprasad
2. soroban
3. MarkFL
4. lfdahl
5. Pranav

Honorable mention goes to both magneto and mente oscura for finding two of the correct $x$ values but missing the other two $x$ values.

soroban's solution:

Spoiler

We have: .$$(2^3)^{(1+|\cos x| + |\cos^2\!x| + |\cos^3\!x| + \cdots)} \:=\: (2^2)^3$$

. . . . . . . . $$2^{3(1+|\cos x| + |\cos^2\!x + |\cos^3\!x| + \cdots)} \:=\:2^6$$

Hence: .$$3(1+|\cos x| + |\cos^2\!x| + |\cos^3\!x| + \cdots) \;=\;6$$

. . . . . . . $$1 + |\cos x| + |\cos^2\!x| + |\cos^3\!x| + \cdots \:=\:2$$We have an infinite series with first term $$a = 1$$, common ratio $$r = |\cos x|$$

Its sum is: .$$\frac{1}{1-|\cos x|}$$We have: .$$\frac{1}{1-|\cos x|} \:=\:2 \quad\Rightarrow\quad 1-|\cos x| \:=\:\tfrac{1}{2}$$

. . . . . . . . $$|\cos x| \:=\:\tfrac{1}{2}$$Therefore: .$$x \;=\;\pm\tfrac{\pi}{3},\;\pm\tfrac{2\pi}{3}$$