The number of solutions of the equation sec x | vedclass

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(C) The given equation is $\sec x = 1 + \cos x + \cos^2 x + \dots \infty$.This is an infinite geometric series with first term $a = 1$ and common rati

Vedclass · Accepted Answer

$100$

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(C) The given equation is $\sec x = 1 + \cos x + \cos^2 x + \dots \infty$.This is an infinite geometric series with first term $a = 1$ and common ratio $r = \cos x$.The sum of an infinite geometric series is given by $S = \frac{a}{1-r}$,provided $|r| Thus,$\sec x = \frac{1}{1 - \cos x}$.This implies $\frac{1}{\cos x} = \frac{1}{1 - \cos x}$,which simplifies to $1 - \cos x = \cos x$.Therefore,$2 \cos x = 1$,or $\cos x = \frac{1}{2}$.In the interval $[0, 2\pi]$,the solutions are $x = \frac{\pi}{3}$ and $x = \frac{5\pi}{3}$.The interval $[-50\pi, 50\pi]$ has a length of $100\pi$,which contains $50$ full periods of $2\pi$.Since there are $2$ solutions in each period of $2\pi$,the total number of solutions is $50 	imes 2 = 100$.

The number of solutions of the equation $\sec x = 1 + \cos x + \cos^2 x + \dots \infty$ in the interval $x \in [-50 \pi, 50 \pi]$ is -

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