Given $\tan^2 2^\circ +\tan^2 4^\circ + \cdots + \tan^2 88^\circ=a$, find the sum $$\sum_{x=1}^{89} \tan^2 x+\cot^2 x$$ in terms of $a$. Since, the given $a$ has only even terms, I am not able to get the odd terms. If I add and subtract these terms, I am unable to manipulate it. Thanks
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Can you better specify what $a$ is? the terms $1^\circ$ and $4^\circ$ are not sufficient to guess. – egreg Aug 10 '15 at 16:49
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could you clarify your question a little bit by showing the whole sum that gives $a$. You mentioned that there are even terms but I see the sum jumps from $1$ to $4$. Is it $tan((2k)^2)$? – Oussama Boussif Aug 10 '15 at 16:49
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Can you use the relationship $\cot(90-x)=\tan x$ in some way? – David Quinn Aug 10 '15 at 16:55
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Sorry, wrote the correct question now. – Aug 10 '15 at 17:10
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I've deleted my answer. I may edit it and then un-delete it. ${}\qquad{}$ – Michael Hardy Aug 10 '15 at 17:26
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2One should note that $\displaystyle \sum_{x=1}^{89} \cot^2 x^\circ = \sum_{x=1}^{89} \tan^2 (90 - x)^\circ = \sum_{y=89}^1 \tan^2 y^\circ $, so the second sum is the same as the first. ${}\qquad{}$ – Michael Hardy Aug 10 '15 at 17:30
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Related : http://math.stackexchange.com/questions/951522/trig-sum-tan-21-circ-tan-22-circ-tan2-89-circ ??? – lab bhattacharjee Aug 11 '15 at 04:42