How to solve the equation sin 3x=sin x, if 0

We have to solve sin 3x = sin x for 0 < x <
pi.

Use sin(A + B) = sin A cos B + cos A sin
B

sin 3x = sin (2x +
x)

=> sin 2x * cos x + cos 2x * sin
x

=> (sin x * cos x + sin x* cos x) cos x + (cos x *
cos x - sin x * sin x)*sin x

=> 2* sin x * (cos x)^2
+ (cos x)^2*sin x - (sin x)^3

=> 3*sin x * (cos x)^2
- (sin x)^3

3*sin x * (cos x)^2 - (sin x)^3 = sin
x

=> 3*(cos x)^2 - (sin x)^2 =
1

=> 3 -  3*(sin x)^2 - (sin x)^2 =
1

=> 3 -  4*(sin x)^2 =
1

=> 4*(sin x)^2 =
2

=> (sin x)^2 =
1/2

=> sin x= 1/sqrt
2

x = arc sin (1/sqrt
2)

=> x =
pi/4

The required solution is x =
pi/4