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Question Number 56282 by Tawa1 last updated on 13/Mar/19

Commented by mr W last updated on 13/Mar/19

$${\sin }^{2}1°+{\sin }^{2}2°+...+{\sin }^{2}44°+{\sin }^{2}45°+{\sin }^{2}46°+...+{\sin }^{2}88°+{\sin }^{2}89°+{\sin }^{2}90°$$$$\mathrm{\%}{\sin }^{2}1°+{\sin }^{2}2°+...+{\sin }^{2}44°+{\sin }^{2}45°+{\cos }^{2}44°+...+{\cos }^{2}2°+{\cos }^{2}1°+{\sin }^{2}90°$$$$=({\sin }^{2}1°+{\cos }^{2}1)°+({\sin }^{2}2°+{\cos }^{2}2°)+...+({\sin }^{2}44°+{\cos }^{2}44°)+{\sin }^{2}45°+{\sin }^{2}90°$$$$=({\sin }^{2}1°+{\cos }^{2}1°)+({\sin }^{2}2°+{\cos }^{2}2°)+...+({\sin }^{2}44°+{\cos }^{2}44°)+{\sin }^{2}45°+{\sin }^{2}90°$$$$=\left(1\right)+\left(1\right)+...+\left(1\right)+{\sin }^{2}45°+{\sin }^{2}90°$$$$=44+\frac{1}{2}+1$$$$=45.5$$

Commented by Tawa1 last updated on 13/Mar/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Commented by Tawa1 last updated on 13/Mar/19

$$\mathrm{I}\mathrm{want}\mathrm{to}\mathrm{also}\mathrm{understand}\mathrm{how}\mathrm{he}\mathrm{solve}\mathrm{using}\mathrm{complex}\mathrm{numbers}.\mathrm{please}.$$

Commented by Tawa1 last updated on 13/Mar/19

$$\mathrm{And}\mathrm{how}\mathrm{he}\mathrm{got}\mathrm{this}\mathrm{step}\mathrm{in}\mathrm{his}\mathrm{solution}:\left(\frac{1-{\mathrm{e}}^{\mathrm{i}(\mathrm{n}+1)\frac{\pi }{90}}}{1-{\mathrm{e}}^{\frac{\mathrm{i}\pi }{90}}}\right)$$

Commented by malwaan last updated on 14/Mar/19

$$veryeasy$$$$thankyousomuch$$

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