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![[Post New]](/templates/default/images/icon_minipost_new.gif) 2 Apr 2008 12:04:36 IST
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[1 ] [n ] tan r tan(r+1)=cot tan(n+1)-n-1
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2 Apr 2008 12:26:44 IST
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![\tan((r+1)\alpha) \tan(r\alpha) =\frac{\tan((r+1)\alpha)-\tan(r\alpha)}{\tan(\alpha)} - 1 \\ \\ \Sigma \frac{\tan((r+1)\alpha)-\tan(r\alpha)}{\tan(\alpha)} - 1 = \cot(\alpha) [ \tan(2\alpha)-\tan(\alpha)+\tan(3\alpha)-\tan(2\alpha)+.....] - n \\ \\ \Longrightarrow \cot(\alpha)*[\tan((n+1)\alpha)-\tan(\alpha)] - n](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/2/6/7/26752ff0326f31275b7ec14b6a64db758a8c8ace.gif)
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this reply: 5 points
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so telescopic cancellations lead us to...... HENCE PROVED......... 
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this reply: 10 points
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in 2 votes ) [?]
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Moderation Team
![[Up]](/templates/default/images/icon_up.gif "[Up]"){kind=icon}
![tan\alpha\;=\;tan[(r+1)\alpha-r\alpha]](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/7/f/2/7f22cf56ac3dbaf56d0eee3b6b9a00441a47476e.gif)
![tan(r+1)\alpha tanr\alpha\;=\;cot\alpha[tan(r+1)\alpha-tanr\alpha]-1](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/6/0/b/60b3e7aae191fbd8eb98134d0d2be70c571383ec.gif)
![\sum_{r=1}^{n}tan(r+1)\alpha tanr\alpha\;=\;\sum_{r=1}^{n}cot\alpha[tan(r+1)\alpha-tanr\alpha]-1](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/1/5/a/15a7b8fcd8f4c517bfc0cd2d1d4880532e0ef3ec.gif)
![=cot\alpha[tan2\alpha-tan\alpha]-1\\+cot\alpha[tan3\alpha -tan2\alpha]-1\\...................\\cot\alpha[tan(n+1)\alpha -tann\alpha]-1](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/1/1/9/119e40ace5c40a545c169aaed8a1446d82b92026.gif)
