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Question Number 147810 by mathdanisur last updated on 23/Jul/21

cos(x) ∙ cos(3x)=cos(5x) ∙ cos(7x)  x = ?

$$\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\:\centerdot\:\boldsymbol{{cos}}\left(\mathrm{3}\boldsymbol{{x}}\right)=\boldsymbol{{cos}}\left(\mathrm{5}\boldsymbol{{x}}\right)\:\centerdot\:\boldsymbol{{cos}}\left(\mathrm{7}\boldsymbol{{x}}\right) \\ $$$$\boldsymbol{{x}}\:=\:? \\ $$

Answered by gsk2684 last updated on 23/Jul/21

2 cos x cos 3x = 2 cos 5x cos 7x  cos (3x+x)+cos (3x−x)=cos (7x+5x)+cos (7x−5x)  cos 4x=cos 12x  12x=2nπ±4x, n∈Z  16x=2nπ or 8x=2nπ  x=((nπ)/8) or ((nπ)/4)

$$\mathrm{2}\:\mathrm{cos}\:{x}\:\mathrm{cos}\:\mathrm{3}{x}\:=\:\mathrm{2}\:\mathrm{cos}\:\mathrm{5}{x}\:\mathrm{cos}\:\mathrm{7}{x} \\ $$$$\mathrm{cos}\:\left(\mathrm{3}{x}+{x}\right)+\mathrm{cos}\:\left(\mathrm{3}{x}−{x}\right)=\mathrm{cos}\:\left(\mathrm{7}{x}+\mathrm{5}{x}\right)+\mathrm{cos}\:\left(\mathrm{7}{x}−\mathrm{5}{x}\right) \\ $$$$\mathrm{cos}\:\mathrm{4}{x}=\mathrm{cos}\:\mathrm{12}{x} \\ $$$$\mathrm{12}{x}=\mathrm{2}{n}\pi\pm\mathrm{4}{x},\:{n}\in{Z} \\ $$$$\mathrm{16}{x}=\mathrm{2}{n}\pi\:{or}\:\mathrm{8}{x}=\mathrm{2}{n}\pi \\ $$$${x}=\frac{{n}\pi}{\mathrm{8}}\:{or}\:\frac{{n}\pi}{\mathrm{4}} \\ $$

Commented by mathdanisur last updated on 23/Jul/21

Thank you Sir

$${Thank}\:{you}\:{Sir} \\ $$

Commented by peter frank last updated on 24/Jul/21

thank you

$${thank}\:{you} \\ $$

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