Question

Find the values of $xϵ(-\pi ,\pi )$ which satisfy the equation $8^{(+|\cos x|+|{\cos }^{2}x|+|{\cos }^{3}x|+......)}=4^3$ (IIT-JEE, 1984)

Answer 1

The given equation is $8^{(+|\cos x|+|{\cos }^{2}x|+|{\cos }^{3}x|+......)}=4^3$

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

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

$\Rightarrow 1+|\cos x|+|{\cos }^{2}x|+|{\cos }^{3}x|+......=2$

$\Rightarrow \frac{1}{1-|\cos x|}=2$

$\Rightarrow |\cos x|=1/2$

$\Rightarrow |\cos x|=\frac{1}{2}$

$\Rightarrow x=\pi /3,-\pi /3,2\pi /3,-2\pi /3,......$

The values of $xϵ(-\pi ,\pi )$ are $\pm \pi /3,\pm 2\pi /3$