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Question Number 115248 by aye48 last updated on 24/Sep/20

Express cos 4x interm of cos x.

$$\mathrm{Express}\:\mathrm{cos}\:\mathrm{4x}\:\mathrm{interm}\:\mathrm{of}\:\mathrm{cos}\:\mathrm{x}. \\ $$

Answered by Olaf last updated on 24/Sep/20

cos4x = 2cos^2 2x−1  cos4x = 2(2cos^2 x−1)^2 −1  cos4x = 2(4cos^4 x−4cos^2 x+1)−1  cos4x = 8cos^4 x−8cos^2 x+1

$$\mathrm{cos4}{x}\:=\:\mathrm{2cos}^{\mathrm{2}} \mathrm{2}{x}−\mathrm{1} \\ $$$$\mathrm{cos4}{x}\:=\:\mathrm{2}\left(\mathrm{2cos}^{\mathrm{2}} {x}−\mathrm{1}\right)^{\mathrm{2}} −\mathrm{1} \\ $$$$\mathrm{cos4}{x}\:=\:\mathrm{2}\left(\mathrm{4cos}^{\mathrm{4}} {x}−\mathrm{4cos}^{\mathrm{2}} {x}+\mathrm{1}\right)−\mathrm{1} \\ $$$$\mathrm{cos4}{x}\:=\:\mathrm{8cos}^{\mathrm{4}} {x}−\mathrm{8cos}^{\mathrm{2}} {x}+\mathrm{1} \\ $$$$ \\ $$

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