Question

Find all values of $x\ne 0$ in $(-\pi ,\pi )$ satisfying the equation $8^{1+\cos x+{\cos }^{2}x+...}=4^3$

Answer 1

Given $8^{1+\cos x+{\cos }^{2}x+........}=4^3$

$\Rightarrow 8^{\frac{1}{1-\cos x}}=4^3$ [$1+\cos x+{\cos }^{2}x+................$ are in infinite $G.P$]

$\Rightarrow [2^{3\left(\frac{1}{1-\cos }\right)}=(2^2{)}^3$ $[SumofinfiniteG.P=\frac{a}{1-r}]$

$2^{\frac{3}{1-\cos x}}=2^6$

$\Rightarrow \frac{3}{1-\cos x}=6\Rightarrow \cos x=\frac{1}{2}$

$x=\frac{-\pi }{3}$ and $\frac{\pi }{3}$