Question 1178306

{{{sin (4x) = 4sin (x) cos^3(x) - 4cos (x) sin^3(x)}}}

Repeatedly use the Double-Angle Identities as needed, and then multiply out the products. (Simplify completely at each step.)


{{{sin (4x) = 4sin (x) cos^3(x) - 4cos (x) sin^3(x)}}}........{{{cos^3(x)=(1/4) (3cos(x) + cos(3 x))}}} and {{{sin^3(x)=(1/4 )(3 sin(x) - sin(3 x))}}}


{{{sin (4x) = 4sin (x) (1/4) (3cos(x) + cos(3 x))  - 4cos (x) (1/4 )(3sin(x) - sin(3 x))}}}


{{{sin (4x) = sin (x) (3cos(x) + cos(3 x))  - cos (x) (3sin(x) - sin(3 x))}}}


{{{sin (4x) =  3sin (x)cos(x) + sin (x)cos(3 x)  - 3sin(x)cos (x)  + cos (x) sin(3 x)}}}


{{{sin (4x) =  sin (x)cos(3 x))  + cos (x) sin(3 x))}}}....{{{cos(3 x)=cos(x) (2 cos(2 x) - 1)}}}, and {{{sin(3 x)=sin(x) (2 cos(2 x) + 1)}}}


{{{sin (4x) =  sin (x)cos(x) (2 cos(2 x) - 1) + cos (x) sin(x) (2 cos(2 x) + 1)}}}


{{{sin (4x) =  sin (x)cos(x) (2 cos(2 x) - 1+ 2 cos(2 x) + 1)}}}


{{{sin (4x) =  sin (x)cos(x) (4 cos(2x) )}}}....{{{sin (x)cos(x)=(1/2) sin(2 x)}}}


{{{sin (4x) =  (1/cross(2)) sin(2 x) (cross(4 )2cos(2x) )}}}


{{{sin (4x) =  sin(2x)(2cos(2x))}}}


{{{sin (4x) = 2sin(2x)cos(2x) }}} 


{{{sin (4x) = sin(4x) }}}