Question

sin(7x)-sin(5x)=sin(x)

Answer 1

\[\sin 7x-\sin 5x=\sin x\]
\[\sin C-\sin D=2 \cos \frac{C+D }{ 2 }\sin \frac{ C-D }{ 2 }\]

Answer 2

\[2 \cos \frac{ 7x+5x }{ 2 }\sin \frac{7x-5x }{2 }-\sin x=0\]
\[\sin x \left( 2\cos 6x-1 \right)=0\]

Answer 3

\[Either \sin x=0=\sin n \pi ,x=n \pi ,n is an integer.\]

Answer 4

\[\cos 6x=\frac{ 1 }{2 }=\cos \frac{ \pi }{3 },\cos \frac{ -\pi }{3 }=\cos \left( 2n \pi \pm \frac{ \pi }{3 } \right)\]
\[6x=2n \pi \pm \frac{ \pi }{ 3 },x=\left( 2n \pm \frac{ 1 }{ 3 } \right)\pi ,n=0,1,2,3,4,5\]