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How to find the solutions of this trigonometric equation $$\sum_{m=1}^6 cosec(\theta + \frac{(m-1)\pi}{4}) cosec(\theta + \frac{m\pi}{4}) =4\sqrt{2}$$ if $0<\theta<\pi/2$.

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HINT:

$\csc\left(\theta+\dfrac{(m-1)\pi}4\right)\cdot\csc\left(\theta+\dfrac{m\pi}4\right)=\dfrac1{\sin\left(\theta+\dfrac{(m-1)\pi}4\right)\cdot\sin\left(\theta+\dfrac{m\pi}4\right)}$

$\dfrac{\sin(A-B)}{\sin A\sin B}=\dfrac{\sin A\cos B-\cos A\sin B}{\sin A\sin B}=\cot B-\cot A$

Here $A=\theta+\dfrac{m\pi}4$ and $B=\theta+\dfrac{(m-1)\pi}4, A-B=\dfrac\pi4$

$\implies\dfrac1{\sin\left(\theta+\dfrac{(m-1)\pi}4\right)\cdot\sin\left(\theta+\dfrac{m\pi}4\right)}$

$=\dfrac1{\sin\dfrac\pi4}\cdot\dfrac{\sin\left\{\theta+\dfrac{m\pi}4-\left(\theta+\dfrac{(m-1)\pi}4\right)\right\}}{\sin\left(\theta+\dfrac{(m-1)\pi}4\right)\cdot\sin\left(\theta+\dfrac{m\pi}4\right)}$

$=\dfrac{\cot\left(\theta+\dfrac{(m-1)\pi}4\right)-\cot\left(\theta+\dfrac{m\pi}4\right)}{\sin\dfrac\pi4}$

Set $m=1,2,3,4,5,6$ and add and finally simplify.

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  • $\begingroup$ @ResearchEngineer, Could u follow the hint? $\endgroup$ – lab bhattacharjee Apr 2 '17 at 11:17
  • $\begingroup$ @ResearchEngineer, put $m=1$ to $6$ to identify the surviving terms. $\endgroup$ – lab bhattacharjee Apr 2 '17 at 11:29
  • $\begingroup$ @ResearchEngineer, Please check the updated answer. $\endgroup$ – lab bhattacharjee Apr 2 '17 at 14:46
  • $\begingroup$ @ResearchEngineer, If $$f(m)=g(m-1)-g(m),$$ can you prove $$\sum_{m=1}^n f(m)=g(0)-g(n)$$ for integer $n\ge1$ $\endgroup$ – lab bhattacharjee Apr 2 '17 at 17:51

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