The period of frac{sin x}{cos 3x} + frac{sin | vedclass

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(D) Let $f(x) = \frac{\sin x}{\cos 3x} + \frac{\sin 3x}{\cos 9x} + \frac{\sin 9x}{\cos 27x} + \frac{\sin 27x}{\cos 81x}$.Using the identity $	an 	he

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$\pi$

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(D) Let $f(x) = \frac{\sin x}{\cos 3x} + \frac{\sin 3x}{\cos 9x} + \frac{\sin 9x}{\cos 27x} + \frac{\sin 27x}{\cos 81x}$.Using the identity $	an 	heta - 	an \phi = \frac{\sin(	heta - \phi)}{\cos 	heta \cos \phi}$,we can rewrite each term:$\frac{\sin x}{\cos 3x} = \frac{1}{2} \frac{\sin(3x - x)}{\cos 3x \cos x} = \frac{1}{2}(	an 3x - 	an x)$.Similarly,$\frac{\sin 3x}{\cos 9x} = \frac{1}{2}(	an 9x - 	an 3x)$,$\frac{\sin 9x}{\cos 27x} = \frac{1}{2}(	an 27x - 	an 9x)$,$\frac{\sin 27x}{\cos 81x} = \frac{1}{2}(	an 81x - 	an 27x)$.Summing these,we get $f(x) = \frac{1}{2}(	an 81x - 	an x)$.The period of $	an(kx)$ is $\frac{\pi}{|k|}$.The period of $	an 81x$ is $\frac{\pi}{81}$ and the period of $	an x$ is $\pi$.The period of the sum is the $L$.$C$.$M$. of the individual periods: $	ext{L.C.M.}\left(\frac{\pi}{81}, \piight) = \pi$.

The period of $\frac{\sin x}{\cos 3x} + \frac{\sin 3x}{\cos 9x} + \frac{\sin 9x}{\cos 27x} + \frac{\sin 27x}{\cos 81x}$ is

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