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Category: Trigonometry

prove-that-2-2-2-2cos-8-2cos-

Question Number 43543 by peter frank last updated on 11/Sep/18 $${prove}\:{that}\:\:\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\mathrm{2cos}\:\mathrm{8}\theta}}} \\ $$$$\mathrm{2cos}\:\theta \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 12/Sep/18 $$\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\mathrm{2}{cos}\mathrm{8}\theta}}} \\ $$$$=\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}\left(\mathrm{1}+{cos}\mathrm{8}\theta\right)}\:}}…

Question-174556

Question Number 174556 by mnjuly1970 last updated on 04/Aug/22 Commented by infinityaction last updated on 04/Aug/22 $$\:\:\:\:\:\:\:\mathrm{tan}\alpha\:+\mathrm{3}\:=\:\mathrm{2sec}\alpha \\ $$$$\:\:\:\:\:\:\mathrm{3tan}^{\mathrm{2}} \alpha\:−\mathrm{6tan}\alpha\:−\mathrm{5}\:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\mathrm{tan}\alpha\:\:=\:\:\:\frac{\mathrm{6}\:\pm\:\mathrm{4}\sqrt{\mathrm{6}}}{\mathrm{6}}\:\:\:=\:\:\frac{\mathrm{sin}\alpha\:}{\mathrm{cos}\alpha\:}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\frac{\mathrm{sin}\alpha−\mathrm{cos}\alpha\:\:}{\mathrm{sin}\alpha+\mathrm{cos}\alpha\:\:}\:\:=\:\:\frac{\mathrm{4}\sqrt{\mathrm{6}}}{\mathrm{12}+\mathrm{4}\sqrt{\mathrm{6}}}\:=\:\frac{\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{3}}+\sqrt{\mathrm{2}}} \\…

prove-that-tan-2tan2-4tan-4-8cot8-cot-

Question Number 43467 by peter frank last updated on 11/Sep/18 $${prove}\:{th}\mathrm{at}\:\mathrm{tan}\theta+\mathrm{2}{tan}\mathrm{2}\theta+\mathrm{4}{tan}\:\:\:\mathrm{4}\theta \\ $$$$+\mathrm{8cot8}\theta={cot}\theta \\ $$ Answered by ajfour last updated on 11/Sep/18 $$\mathrm{8cot}\:\mathrm{8}\theta+\mathrm{4tan}\:\mathrm{4}\theta+\mathrm{2tan}\:\mathrm{2}\theta+\mathrm{tan}\:\theta \\ $$$$\:\:\:\:=\:\frac{\mathrm{8}\left(\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}}…

1-cos-2-10-1-sin-2-20-1-sin-2-40-1-cos-2-45-

Question Number 174346 by cortano1 last updated on 30/Jul/22 $$\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{cos}\:^{\mathrm{2}} \mathrm{10}°}\:+\frac{\mathrm{1}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{20}°}\:+\frac{\mathrm{1}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{40}°}\:−\frac{\mathrm{1}}{\mathrm{cos}\:^{\mathrm{2}} \mathrm{45}°\:}=? \\ $$ Answered by behi834171 last updated on 30/Jul/22 $$\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \mathrm{10}}=\mathrm{1}+{tg}^{\mathrm{2}}…

probably-cos-nx-2-n-1-cos-n-x-n2-n-3-cos-n-2-n-3-n-2-2-n-5-cos-n-4-x-wow-

Question Number 43268 by Rauny last updated on 09/Sep/18 $$\mathrm{probably},\:\mathrm{cos}\:{nx}=\mathrm{2}^{{n}−\mathrm{1}} \mathrm{cos}^{{n}} \:{x}−{n}\mathrm{2}^{{n}−\mathrm{3}} \mathrm{cos}^{{n}−\mathrm{2}} \: \\ $$$$+\frac{\left({n}−\mathrm{3}\right){n}}{\mathrm{2}}\mathrm{2}^{{n}−\mathrm{5}} \mathrm{cos}^{{n}−\mathrm{4}} \:{x}\ldots \\ $$$$\mathrm{wow} \\ $$ Commented by Rauny…

cos-2x-2cos-2-1-cos-3x-4cos-3-x-3cos-x-cos-4x-8cos-4-x-8cos-2-x-1-cos-5x-16cos-5-x-20cos-3-5cos-x-cos-6x-32cos-6-x-48cos-4-x-18cos-2-x-1-cos-7x-64cos-7-x-112cos-5-x-56cos-3-x-4cos-x-cos-8

Question Number 43267 by Rauny last updated on 09/Sep/18 $$\mathrm{cos}\:\mathrm{2}{x}=\mathrm{2cos}^{\mathrm{2}} −\mathrm{1} \\ $$$$\mathrm{cos}\:\mathrm{3}{x}=\mathrm{4cos}^{\mathrm{3}} \:{x}−\mathrm{3cos}\:{x} \\ $$$$\mathrm{cos}\:\mathrm{4}{x}=\mathrm{8cos}^{\mathrm{4}} \:{x}−\mathrm{8cos}^{\mathrm{2}} \:{x}+\mathrm{1} \\ $$$$\mathrm{cos}\:\mathrm{5}{x}=\mathrm{16cos}^{\mathrm{5}} \:{x}−\mathrm{20cos}^{\mathrm{3}} +\mathrm{5cos}\:{x}\: \\ $$$$\mathrm{cos}\:\mathrm{6}{x}=\mathrm{32cos}^{\mathrm{6}} \:{x}−\mathrm{48cos}^{\mathrm{4}}…