Question

Rewrite sin(x)cos2(x) − sin(x) as an expression containing a single term

Answer 1

Answer: -2sin³x

Explanation:

cos ( x + y ) = cosx*cosy - sin y*sinx (1)

if x = y (1) could be express as: cos (2x) = cosx*cosx -sinx*sinx

cos 2x = cos²x -sin²x

Then sin(x)*cos2(x) − sin(x) ⇒ sinx * ( cos²x - sin²x ) - sinx

multiplying sinx*cos²x -sin³x -sinx

sinx (commom factor sinx (cos²x -sin²x -1) but 1-cos²x = sin²x

or -1 +cos²x = -sin²x ⇒sinx( -sin²x -sin²x) ⇒sinx (-2sin²x)

-2sin³x