cos 15x = sin 5x, for what values of x ?

Dear Student,

Please find below the solution to the asked query :


$$\begin{gathered} \cos 15 \mathrm{x} = \sin 5\mathrm{x} \\ \Rightarrow \cos 15\mathrm{x} = \cos \left(\frac{\mathrm{\pi }}{2} - 5\mathrm{x}\right) \\ \Rightarrow 15\mathrm{x} = 2\mathrm{n\pi }\pm \left(\frac{\mathrm{\pi }}{2} - 5\mathrm{x}\right) \\ \Rightarrow 15\mathrm{x} = 2\mathrm{n\pi } + \frac{\mathrm{\pi }}{2} - 5\mathrm{x} \mathrm{or} 15\mathrm{x} = 2\mathrm{n\pi }-\frac{\mathrm{\pi }}{2}+5\mathrm{x} \\ \Rightarrow 20\mathrm{x} = 2\mathrm{n\pi }+\frac{\mathrm{\pi }}{2} \mathrm{or} 10\mathrm{x} = 2\mathrm{n\pi } - \frac{\mathrm{\pi }}{2} \\ \Rightarrow \mathrm{x} = \frac{\mathrm{n\pi }}{10}+\frac{\mathrm{\pi }}{40} \mathrm{or} \mathrm{x} = \frac{\mathrm{n\pi }}{5} - \frac{\mathrm{\pi }}{20} ; \mathrm{n}\in \mathrm{I} \end{gathered}$$

Hope this would clear your doubt.

Do let us know in case of any further concerns.

Regards