Question

Rewrite with only sin x and cos x.

sin (3x) - cos (x)

Please show answer with work

Answer 1

The main identities we use are:

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

ii)
`sin(2a)=2sin(a)sin(b)`

iii)
`cos(2x)= cos^(2)(x)-sin^(2)(x)`


Thus,

sin (3x)=sin(2x+x)=sin(2x)cos(x) + cos(2x)sin(x)


`=2sin(x)cos(x)cos(x)+[cos^(2)(x)-sin^(2)(x)]sin(x)`


`=2sin(x)cos^(2) (x)+sin(x)cos^(2)(x)-sin^(3)(x)`


`=3sin(x)cos^(2)(x)-sin^(3)(x)`


Answer:
`3sin(x)cos^(2)(x)-sin^(3)(x)`