Finding solution of trigonometric equation sum_(m=1)^6 cosec(theta + ((m-1) pi)/(4)) cosec(theta + (m pi)/(4)) =4 sqrt(2) if 0<theta<pi/2.

Marlene Brooks 2022-10-25 Answered
Finding solution of trigonometric equation
How to find the solutions of this trigonometric equation
$\sum _{m=1}^{6}cosec\left(\theta +\frac{\left(m-1\right)\pi }{4}\right)cosec\left(\theta +\frac{m\pi }{4}\right)=4\sqrt{2}$
if $0<\theta <\pi /2$
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Kaylee Evans
HINT:
$\mathrm{csc}\left(\theta +\frac{\left(m-1\right)\pi }{4}\right)\cdot \mathrm{csc}\left(\theta +\frac{m\pi }{4}\right)=\frac{1}{\mathrm{sin}\left(\theta +\frac{\left(m-1\right)\pi }{4}\right)\cdot \mathrm{sin}\left(\theta +\frac{m\pi }{4}\right)}$
$\frac{\mathrm{sin}\left(A-B\right)}{\mathrm{sin}A\mathrm{sin}B}=\frac{\mathrm{sin}A\mathrm{cos}B-\mathrm{cos}A\mathrm{sin}B}{\mathrm{sin}A\mathrm{sin}B}=\mathrm{cot}B-\mathrm{cot}A$
Here $A=\theta +\frac{m\pi }{4}$ and $B=\theta +\frac{\left(m-1\right)\pi }{4},A-B=\frac{\pi }{4}$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\frac{1}{\mathrm{sin}\left(\theta +\frac{\left(m-1\right)\pi }{4}\right)\cdot \mathrm{sin}\left(\theta +\frac{m\pi }{4}\right)}$
$=\frac{1}{\mathrm{sin}\frac{\pi }{4}}\cdot \frac{\mathrm{sin}\left\{\theta +\frac{m\pi }{4}-\left(\theta +\frac{\left(m-1\right)\pi }{4}\right)\right\}}{\mathrm{sin}\left(\theta +\frac{\left(m-1\right)\pi }{4}\right)\cdot \mathrm{sin}\left(\theta +\frac{m\pi }{4}\right)}$
$=\frac{\mathrm{cot}\left(\theta +\frac{\left(m-1\right)\pi }{4}\right)-\mathrm{cot}\left(\theta +\frac{m\pi }{4}\right)}{\mathrm{sin}\frac{\pi }{4}}$
Set $m=1,2,3,4,5,6$ and add and finally simplify.