Simplify
\(4 \sin x \sin (60^\circ - x) \sin (60^\circ + x)\)
The answer will be a trigonometric function of some simple function of x like "2sinx" or "4cosx".
I think the way to solve this is by using Sum to Product and Product to Sum formulas but whenever i try i get a wrong answer.
+23245 I'm going to use these identities:
1) sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
2) sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
3) sin(3A) = 3sin(A) - 4sin2(A)
4 · sin(x) · sin(60 -x) · sin(60+x)
= 4 · sin(x) · [ sin(60 - x) ] · [ sin(60 + x) ]
= 4 · sin(x) · [ sin(60)cos(x) - cos(60)sin(x) ] · [ sin(60)cos(x) + cos(60)sin(x) ]
= 4 · sin(x) · [ sqrt(3)cos(x)/2 - sin(x)/2 ] · [ sqrt(3)cos(x)/2 + sin(x)/2 ]
= 4 · sin(x) · [ 3cos2(x)/4 - sin2(x)/4 ]
= sin(x) · [ 3cos2(x) - sin2(x) ]
= sin(x) · [ 3( 1 - sin2(x) ) - sin2(x) ]
= sin(x) · [ 3 - 3sin2(x) - sin2(x) ]
= sin(x) · [ 3 - 4sin2(x) ]
= 3sin(x) - 4sin3(x)
= sin(3x)
