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Question Number 43467 by peter frank last updated on 11/Sep/18

$$proveth\mathrm{at}\tan \theta +2tan2\theta +4tan4\theta$$$$+8\cot 8\theta =cot\theta$$

Answered by ajfour last updated on 11/Sep/18

$$8\cot 8\theta +4\tan 4\theta +2\tan 2\theta +\tan \theta$$$$=\frac{8(1-\tan {}^{2}4\theta )}{2\tan 4\theta }+4\tan 4\theta +...$$$$=4\cot 4\theta +2\tan 2\theta +\tan \theta$$$$=\frac{4(1-\tan {}^{2}2\theta )}{2\tan 2\theta }+2\tan 2\theta +\tan \theta$$$$=2\cot 2\theta +\tan \theta$$$$=\frac{2(1-\tan {}^2\theta )}{2\tan \theta }+\tan \theta =\cot \theta .$$

Commented by peter frank last updated on 11/Sep/18

$$thanks$$

Commented by peter frank last updated on 11/Sep/18

$$$$$$\mathrm{sir}\mathrm{the}\mathrm{third}\mathrm{line}\mathrm{am}\mathrm{not}\mathrm{understood}$$$$\mathrm{how}\mathrm{you}\mathrm{get}4\cot 4\theta$$$$$$$$$$

Commented by peter frank last updated on 11/Sep/18

$$$$$$\mathrm{sir}\mathrm{the}\mathrm{third}\mathrm{line}\mathrm{am}\mathrm{not}\mathrm{understood}$$$$\mathrm{how}\mathrm{you}\mathrm{get}4\cot 4\theta$$$$$$$$$$$$$$

Commented by ajfour last updated on 11/Sep/18

$$\frac{8}{2\tan 4\theta }=4\cot 4\theta$$

Answered by tanmay.chaudhury50@gmail.com last updated on 11/Sep/18

$$tan\theta -cot\theta$$$$=tan\theta -\frac{1}{tan\theta }$$$$=\frac{{tan}^2\theta -1}{tan\theta }$$$$=\frac{-2(1-{tan}^2\theta )}{2tan\theta }$$$$=-2cot2\theta$$$$sotan\theta =cot\theta -2cot2\theta$$$$2tan2\theta =2(cot2\theta -2cot4\theta )$$$$4tan4\theta =4(cot4\theta -2cot8\theta )$$$$8cot8\theta =8cot8\theta$$$$nowaddthemRHS=cot\theta$$$$newtricks...noneedtocalculatdtan2\theta ,tan4\theta$$$$cot8\theta etc...$$$$$$$$$$

Commented by peter frank last updated on 11/Sep/18

$$thanks$$

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