please help with verifying the identity: cos(180 degrees-x)=-cosx

Two ways to do this:

1. Use the formula for the cosine of the sum of angles:

cos(a+b) = cos(a)cos(b)-sin(a)sin(b),

where a = 180 degrees and b = -x. Remember that sin(180)=0 and cos(180) = -1.

2. Use some of the rues for graphing functions which you learned in algebra. First, consider the function cos(-x); because the cosine function is symmetric about the y axis the graph of cos(-x) is the same as cos(x). Next, consider that f(180-x) is equivalent to shifting the graph of f(x) to the left by 180. Put these two facts together and you see that the graph of cos(180-x) is the cos(x) graph shifted by 180. This put it exactly "out of phase" with cos(x), and hence cos(180-x) = -cos(x).