Use an Addition or Subtraction Formula to simplify the equat

Use an Addition or Subtraction Formula to simplify the equation. sin(3 theta) cos(theta) - cos(3theta) sin(theta) = Squareroot 3 /2 Find all solutions in the interval [0, 2pi). (Enter your answers as a comma-separated list.)

Solution

I am taking theta as\'x\'

so the given problem becomes sin(3x).cos(x) - cos(3x) .sin(x) = sqrt(3)/2

this is of the form Sin(A).cos(B) - cos(A).sin(B) = sin(A-B)

here A=3x , B=x

sin(3x-x) = sqrt(3)/2

sin(2x) = sqrt(3)/2 ----> Answer for 1st box

now finding solutions

2x = pi/3 , 2pi/3

x = pi/6, pi/3

solutions are =pi/6,pi/3