Use an addition or subtraction formula to simplify the equation.
sin(θ) cos(3θ) + cos(θ) sin(3θ) = 0
find all solutions in the interval [0, 2π). (enter your answers as a comma-separated list.)

Sin (A+B)=sin A cos B+cos A cos B
In this equation:
A=Ф
B=3Ф

Therefore:
sin (Ф) cos(3Ф)+cos (Ф) sin (3Ф)=sin (Ф+3Ф)=sin (4Ф).

Then, we have:
sin (4Ф)=0
4Ф=sin⁻¹ 0=kπ      (k∈Z)    Z=...-2,-1,0,1,2,...
4Ф=kπ
Ф=kπ/4

Ф=Kπ/4    (k∈Z)     Z=...-2,-1,0,1,2...

We can check it out our answer is right.
For example:
if  Z=1; then Ф=π/4
sin(π/4)+cos(3π/4)+cos(π/4)sin(3π/4)=(√2 /2)(-√2/2)+(√2/2)(√2/2)=
=-1/2+1/2=0