pre calc sin(6x)-cos(6x)

Question

Write the expression in terms of sine only.

sin(6x) − cos(6x)

Answer 1

If you take the cosine part of the initial expression and rewrite it as a double angle, cos 2(3x), then by one of the double-angle identities for cosine, we can write cos 2(3x) as 1 - 2sin² (3x). Now, combining this with the first part of the original expression we have sin (6x) - [ 1 - 2sin² (3x)], which simplifies to sin (6x) - 1 + 2sin² (3x).

Answer 2

The double angle formula gives cos(2x) = 1- 2sin²(x). Thus, cos(6x) = cos(2(3x)) = 1- 2sin²(3x). How can you use this to write your expression in terms of only sine?