The number of values of x lying in the interval (−π, π) which satisfy the equation 8^{1+|cos x|+cos²x+|cos³x|+....∞} = 4³ is:
4
The exponent is the infinite geometric series \(1+|\cos x|+\cos^2 x+|\cos^3 x|+\cdots = \sum_{n=0}^{\infty}|\cos x|^n\), since \(|\cos^n x| = |\cos x|^n\).
This series converges to \(\dfrac{1}{1-|\cos x|}\) (for \(|\cos x|<1\)), so the equation becomes \(8^{1/(1-|\cos x|)} = 4^3 = 2^6\).
Since \(8=2^3\), this gives \(2^{3/(1-|\cos x|)} = 2^6\), so \(\dfrac{3}{1-|\cos x|}=6\), hence \(1-|\cos x| = \dfrac{1}{2}\), giving \(|\cos x| = \dfrac{1}{2}\).
In \((-\pi,\pi)\), \(\cos x = \tfrac{1}{2}\) gives \(x = \pm\pi/3\), and \(\cos x = -\tfrac{1}{2}\) gives \(x = \pm 2\pi/3\).
So there are 4 distinct solutions in \((-\pi,\pi)\).
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