For cos x cos 3x sin x sin 3x 0 use an addition or subtrac

For cos x cos 3x - sin x sin 3x = 0, use an addition or subtraction formula to simplify the equation and then find all solution of the equation in the interval [0, pi]. The answer is x_1 = x_2 = x_3 = x_4 = with x_1

Solution

cos x cos 3x - sinx sin 3x=0

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

So we can write the given equation as

cos(x+3x)=0

cos 4x=0

taking inverse on both sides

4x= cos^-1(0)

4x=pi/2,3pi/2,5pi/2,7pi/2

x=pi/8,3pi/8,5pi/8,7pi/8

therefore x1=pi/8

x2=3pi/8

x3=5pi/8

x4=7pi/8