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

To simplify the expression cos(2x+4h) - cos(2x+2h), we can start by using the formula for the difference of two cosines.

The formula states:

cos(a) - cos(b) = -2sin((a+b)/2)sin((a-b)/2)

Comparing this formula with our expression, we have a = 2x+4h and b = 2x+2h.

So, let's substitute these values into the formula:

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

Now, simplify the expression inside the sine functions:

= -2sin((4x+6h)/2)sin((2h)/2)
= -2sin(2x+3h)sin(h)

Hence, the simplified expression for cos(2x+4h) - cos(2x+2h) is -2sin(2x+3h)sin(h).