If $P=\tan(3^{n+1}\theta)-\tan\theta$ and $Q=\sum_{r=0}^n\frac{\sin(3^r\theta)}{\cos(3^{r+1}\theta)}$,then relate $P$ and $Q$


I do not know how to change into telescoping series.



$$\tan3A-\tan A=\dfrac{\sin(3A-A)}{\cos3A\cos A}=\dfrac{2\sin A}{\cos3A}$$

Do you recognize the Telescoping nature?

Related : Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$


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