Rewrite with only sin x and cos x. sin 3x I don't know how to do this. Can you help me?

Question

2 sin x cos^2 x + cos x

2 sin x cos^2 x + sin^3 x
sin x cos^2 x - sin^3 x + cos^3 x
2 cos^2 x sin x + sin x - 2 sin^3 x

 

these are the options so the answer has to be one of these

Answer 1

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

 

           = sinx cos(2x) + cosx sin(2x)

 

           = sinx [1 - 2sin²x] + cosx [2sinxcosx]

 

           = sinx - 2sin³x + 2cos²xsinx

 

          

Answer 2

remember that 

 

sin(a +b) = sin(a) cos(b) + sin(b) cos (a)       call this #1

 

if a and b are both =x then sin (x + x) = 2sinx cosx      call this #2

 

and cos2x = 1 - 2sin²x        call this #3

 

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

 

= sin2x cosx + sinx cos 2x     using #1

 

   = (2cosx sinx)(cos x) + sinx (1 - 2sin²x)    using #2 and #3

 

=2cos²x sinx + sinx - 2sin³x

 

= your fourth choice