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Author: Tinku Tara

Prove-1-2-n-1-a-n-cos-nx-1-a-2-1-2acos-x-a-2-a-lt-1-And-calculate-0-pi-x-2-1-8sin-2-x-dx-5pi-3-36-pi-6-ln-2-2-

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Question Number 139708 by qaz last updated on 30/Apr/21 $${Prove}:\:\:\:\mathrm{1}+\mathrm{2}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}^{{n}} \mathrm{cos}\:\left({nx}\right)=\frac{\mathrm{1}−{a}^{\mathrm{2}} }{\mathrm{1}−\mathrm{2}{a}\mathrm{cos}\:{x}+{a}^{\mathrm{2}} },\:\:\:\:\:\:\left(\mid{a}\mid<\mathrm{1}\right) \\ $$$${And}\:{calculate}::\:\:\int_{\mathrm{0}} ^{\pi} \frac{{x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{8sin}\:^{\mathrm{2}} {x}}{dx}=\frac{\mathrm{5}\pi^{\mathrm{3}} }{\mathrm{36}}−\frac{\pi}{\mathrm{6}}{ln}^{\mathrm{2}} \mathrm{2} \\ $$…

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If-3sin-x-5cos-x-5-find-value-s-of-5sin-x-3cos-x-

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Question Number 139710 by ajfour last updated on 30/Apr/21 $${If}\:\:\mathrm{3sin}\:{x}+\mathrm{5cos}\:{x}=\mathrm{5} \\ $$$${find}\:{value}\left({s}\right)\:{of}\:\:\mathrm{5sin}\:{x}−\mathrm{3cos}\:{x}. \\ $$ Answered by MJS_new last updated on 30/Apr/21 $$\mathrm{3sin}\:{x}\:+\mathrm{5cos}\:{x}\:=\mathrm{5} \\ $$$$\frac{\mathrm{3tan}\:{x}+\mathrm{5}}{\:\sqrt{\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \:{x}}}=\mathrm{5}\:\Rightarrow\:\mathrm{tan}\:{x}\:=\mathrm{0}\vee\mathrm{tan}\:{x}\:=\frac{\mathrm{15}}{\mathrm{8}}…

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