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For $ 0 <\theta<\frac{\pi}{2}$ find the solution of

$$\sum\limits_{m=1}^{6}\csc\left(\theta+\frac{(m-1)\pi}{4}\right)\cdot\csc\left(\theta+\frac{m\pi}{4}\right)=4\sqrt{2}$$

I thought of solving this as the angles form an A.P , But the given sum does not come under any standard type such as the sum of the sines or cosines of the angles in an A.P.So I am unable to proceed further.

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    Is $m$ an integer? How about writing $\csc=\frac{1}{\sin}$ and expanding the angle sums? Just a thought.2011-06-20
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    I modified the sum again.2011-06-20
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    $\csc(\theta+\pi) = -\csc(\theta)$ so there is some cancellation...2011-06-20
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    @gedgar here it is pi/4 instead of pi.2011-06-20
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    @GEdgar: so the $m=1$ term is the same as the $m=5$ term, and also for $2$ and $6$2011-06-20

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