George Michael

2022-04-06

Finding sum of $\sum _{\left\{x=1\right\}}^{17}\frac{1}{1+{\mathrm{cot}}^{4}\left({10}^{\circ }x\right)}$

embemiaEffoset4rs

Hint
$\sum _{\left\{x=1\right\}}^{17}\frac{1}{1+{\mathrm{cot}}^{4}\left({10}^{\circ }x\right)}=\sum _{\left\{x=1\right\}}^{17}\frac{{\mathrm{sin}}^{4}\left({10}^{\circ }x\right)}{{\mathrm{sin}}^{4}\left({10}^{\circ }x\right)+{\mathrm{cos}}^{4}\left({10}^{\circ }x\right)}$
Hint 2
$\sum _{\left\{x=1\right\}}^{8}\frac{{\mathrm{sin}}^{4}\left({10}^{\circ }x\right)}{{\mathrm{sin}}^{4}\left({10}^{\circ }x\right)+{\mathrm{cos}}^{4}\left({10}^{\circ }x\right)}=\sum _{\left\{x=1\right\}}^{8}\frac{{\mathrm{cos}}^{4}\left({90}^{\circ }-{10}^{\circ }x\right)}{{\mathrm{cos}}^{4}\left({90}^{\circ }-{10}^{\circ }x\right)+{\mathrm{sin}}^{4}\left({90}^{\circ }-{10}^{\circ }x\right)}$
Hint 3
What is
$\sum _{\left\{x=1\right\}}^{8}\frac{{\mathrm{sin}}^{4}\left({10}^{\circ }x\right)}{{\mathrm{sin}}^{4}\left({10}^{\circ }x\right)+{\mathrm{cos}}^{4}\left({10}^{\circ }x\right)}+\sum _{\left\{x=1\right\}}^{8}\frac{{\mathrm{cos}}^{4}\left({10}^{\circ }x\right)}{{\mathrm{sin}}^{4}\left({10}^{\circ }x\right)+{\mathrm{cos}}^{4}\left({10}^{\circ }x\right)}$
Hint 4: Use the same trick for
$\sum _{\left\{x=10\right\}}^{17}\frac{{\mathrm{sin}}^{4}\left({10}^{\circ }x\right)}{{\mathrm{sin}}^{4}\left({10}^{\circ }x\right)+{\mathrm{cos}}^{4}\left({10}^{\circ }x\right)}$

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