Question Number 105692 by bramlex last updated on 31/Jul/20
$$\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{5}{x}\right)+\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{7}{x}\right)=\mathrm{1} \\ $$
Answered by john santu last updated on 31/Jul/20
$$\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{5}{x}\right)+\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{7}{x}\right)=\mathrm{1} \\ $$$$\left\{\mathrm{cos}\:\left(\mathrm{5}{x}\right)+\mathrm{cos}\:\left(\mathrm{7}{x}\right)\right\}\left\{\mathrm{cos}\:\left(\mathrm{7}{x}\right)−\mathrm{cos}\left(\mathrm{5}{x}\right)\right\}=\mathrm{0} \\ $$$$\left\{\mathrm{2cos}\:\left(\mathrm{6}{x}\right)\mathrm{cos}\:\left({x}\right)\right\}\left\{−\mathrm{2sin}\:\left(\mathrm{6}{x}\right)\mathrm{sin}\left({x}\right)\right\}=\mathrm{0} \\ $$$$\mathrm{sin}\:\left(\mathrm{12}{x}\right)\mathrm{sin}\:\left(\mathrm{2}{x}\right)=\mathrm{0} \\ $$$$\begin{cases}{\mathrm{sin}\:\left(\mathrm{12}{x}\right)=\mathrm{0}}\\{\mathrm{sin}\:\left(\mathrm{2}{x}\right)=\mathrm{0}}\end{cases} \\ $$$$\begin{cases}{\mathrm{12}{x}={n}\pi\rightarrow{x}=\frac{{n}\pi}{\mathrm{12}}}\\{\mathrm{2}{x}={n}\pi\rightarrow{x}=\frac{{n}\pi}{\mathrm{2}}}\end{cases}\:\left({JS}\:\spadesuit\Box\right) \\ $$