Solve the equation sinx = sin5x

To solve this equation, we'll move the term from the right
side, to the left side:

sinx - sin5x =
0

As we can see, it is a subtraction of 2 trigonometric
function of the same kind and it is transforming into a
product.

2 cos [(x+5x)/2]*sin
[(x-5x)/2]=0

2 cos 3x* sin
(-2x)=0

From this product of 2 factors, one or the other
factor is zero.

cos 3x=0

This
is an elementary equation:

3x = +/-arccos 0 +
2*k*pi

3x=+/- (pi/2)+
2*k*pi

x=+/- (pi/6)+ 2*k*pi/3, where k is an integer
number.

We'll solve the second elementary
equation:

sin (-2x)=0

-sin
(2x)=0

sin 2x=0

2x=(-1)^k
arcsin 0 +
k*pi

2x=k*pi

x=k*pi/2

The
set of solutions:

S={+/- (pi/6)+
2*k*pi/3}or{k*pi/2}