Verified answer

The angles you are given are given in radians... if you want to convert to degrees do this

degrees = x radians * 180 degrees / pi radians

so

In the case of theta = π/3

theta = π/3 * 180/π

= 180/3

= 60 degrees

You can do the same for pi/6 and you should get 30 degrees.

Sine Addition Formula

sin(a+b) = sin(a)cos(b) + sin(b)cos(a)

Cosine Addition Formula

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

So

sin(x+(π/3)) = sin(x)cos(π/3) + sin(π/3)cos(x)

cos(x+(π /6))= cos(x)cos(π /6) - sin(x)sin(π /6)

Simplifying

sin(x+(π/3)) = sin(x)cos(60) + sin(60)cos(x)

= 1/2 * sin(x) + sqrt(3)/2 cos(x)

cos(x+(π /6))= cos(x)cos(30) - sin(x)sin(30)

= sqrt(3)/2 cos(x) - 1/2sin(x)

so

sin(x+(π/3))-cos(x+(π /6))= 1/2 * sin(x) + sqrt(3)/2 cos(x) - [sqrt(3)/2 cos(x) - 1/2sin(x)]

= 1/2 * sin(x) + sqrt(3)/2 cos(x) - sqrt(3)/2 cos(x) + 1/2sin(x)

= 1/2 sin(x) + 1/2 sin(x)

= sin(x)