[tex]\displaystyle \\
\texttt{The equation has three solutions.} \\ \\
S1: \\
\sin(7x) = \sin(5x) \\
7x = 5x \\
7x-5x=0 \\
2x=0 \\
\boxed{x_1 = 0^o} \\
Check: \\
\sin(7\cdot 0)= \sin(0) = 0~~and~~\sin(5\cdot 0)= \sin(0) = 0
[/tex]
[tex]\displaystyle S2: \\
\sin(7x) = \sin(5x) \\
\sin(180-\alpha)=\sin(\alpha) ~~\Longrightarrow~~ \sin(90+\beta)=\sin(90-\beta) \\
\Longrightarrow~~ \frac{7x + 5x}{2} = 90^o \\ \\
\frac{12x}{2} = 90^o \\ \\
6x = 90^o \\ \\
x_2 = \frac{90}{6} \\ \\
\boxed{x_2 = 15^o }\\ \\
Check: \\
\sin(7\cdot 15^o) = \sin(105^o)=\sin(180^o-105^o)=\sin(75^o)=\sin(5\cdot 15^o)
[/tex]
[tex]\displaystyle S3: \\
\sin(7x) = \sin(5x) \\
\sin(180+\alpha)=\sin(360-\alpha) ~~\Longrightarrow~~ \sin(270+\beta)=\sin(270-\beta) \\
\Longrightarrow~~ \frac{7x + 5x}{2} = 270^o \\ \\
\frac{12x}{2} = 270^o \\ \\
6x = 270^o \\ \\
x_3 = \frac{270}{6} \\ \\
\boxed{x_3 = 45^o }\\ \\
Check: \\
\sin(7\cdot 45^o) = \sin(315^o)=\sin(360^o-45^o)=\sin(180+45^o)= \\ = \sin(225^o) = \sin(5 \cdot 45^0) [/tex]
