Question

Find all solutions to cosx cos3 x-sinx sine3x= 0 on the interval {0,2pi}

Answer 1

$\bf \textit{Sum and Difference Identities} \\ \quad \\ sin({{ \alpha}} + {{ \beta}})=sin({{ \alpha}})cos({{ \beta}}) + cos({{ \alpha}})sin({{ \beta}}) \\ \quad \\ sin({{ \alpha}} - {{ \beta}})=sin({{ \alpha}})cos({{ \beta}})- cos({{ \alpha}})sin({{ \beta}}) \\ \quad \\ \boxed{cos({{ \alpha}} + {{ \beta}})= cos({{ \alpha}})cos({{ \beta}})- sin({{ \alpha}})sin({{ \beta}})} \\ \quad \\ cos({{ \alpha}} - {{ \beta}})= cos({{ \alpha}})cos({{ \beta}}) + sin({{ \alpha}})sin({{ \beta}})\\\\ ------------------------$

$\bf cos(x)cos(3x)-sin(x)sin(3x)=0\implies cos(x+3x)=0 \\\\\\ cos(4x)=0\implies cos^{-1}[cos(4x)]=cos^{-1}(0)\implies 4x=cos^{-1}(0) \\\\\\ 4x= \begin{cases} \frac{\pi }{2}\\\\ \frac{3\pi }{2} \end{cases}\implies \begin{cases} 4x=\cfrac{\pi }{2}\implies &\measuredangle x=\cfrac{\pi }{8}\\\\ 4x=\cfrac{3\pi }{2}\implies &\measuredangle x=\cfrac{3\pi }{8} \end{cases}$