Question

Rewrite with only sin x and cos x.

sin (3x) - cos (x)

Please guide me with showing work because the process of substituting is what I'm having difficulties with.

Answer 1

sin (3 x) - cos (x)=
= sin (2x + x) - cos (x)=
= sin 2x * cos x + cos 2 x * sin x - cos x = ( using additional formulas )
= 2 sin x cos² x + ( cos² x - sin² x ) sin x - cos x =
= 2 sin x cos² x + ( 1 - sin² x - sin² x ) sin x - cos x =
= 2 sin x cos² x + ( 1 - 2 sin² x ) sin x - cos x =
= 2 sin x cos² x + sin x - 2 sin³ x - cos x
Answer: B )