The main identities we use are:
i) [tex]sin(a+b)=sin(a)cos(b)+sin(b)cos(a)[/tex]
ii) [tex]sin(2a)=2sin(a)sin(b)[/tex]
iii) [tex]cos(2x)= cos^{2}(x)-sin^{2}(x) [/tex]
Thus,
sin (3x)=sin(2x+x)=sin(2x)cos(x) + cos(2x)sin(x)
[tex]=2sin(x)cos(x)cos(x)+[cos^{2}(x)-sin^{2}(x)]sin(x)[/tex]
[tex]=2sin(x)cos^{2} (x)+sin(x)cos^{2}(x)-sin^{3}(x)[/tex]
[tex]=3sin(x)cos^{2}(x)-sin^{3}(x)[/tex]
Answer: [tex]3sin(x)cos^{2}(x)-sin^{3}(x)[/tex]
