cos(2x+4h)-cos(2x+2h)=?

To simplify the expression cos(2x+4h) - cos(2x+2h), we can use the trigonometric identity called the cosine of the difference of angles:

cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Let's use this identity to simplify the expression:

cos(2x+4h) - cos(2x+2h) = [cos(2x)cos(4h) + sin(2x)sin(4h)] - [cos(2x)cos(2h) + sin(2x)sin(2h)]

Now, let's expand these expressions:

= cos(2x)cos(4h) + sin(2x)sin(4h) - cos(2x)cos(2h) - sin(2x)sin(2h)

Since the terms cos(2x)cos(4h) and -cos(2x)cos(2h) have the same cosine value, and the terms sin(2x)sin(4h) and -sin(2x)sin(2h) have the same sine value, we can group these together:

= cos(2x)cos(4h) - cos(2x)cos(2h) + sin(2x)sin(4h) - sin(2x)sin(2h)

Now, let's apply another trigonometric identity called the difference of angles formula for cosine:

cos(A - B) = cos(A)cos(B) - sin(A)sin(B)

Using this identity, we can simplify the expression further:

= cos(2x)(cos(4h) - cos(2h)) + sin(2x)(sin(4h) - sin(2h))

And there you have it! The simplified form of the expression cos(2x+4h) - cos(2x+2h) is cos(2x)(cos(4h) - cos(2h)) + sin(2x)(sin(4h) - sin(2h)).