I will use d for degrees.
cos80dcos20d - sin80dsin20d =
I got -0.173648178
Is this correct?
So to work out the value of you can simply use a calculator and find instead.
Or, of course, you can do it the long way and find the separate values of and , and then multiply and add. You should get the same answer whichever way you choose.
Do you mean how do you find without using a calculator? If so, I'm not sure that you can. There are some angles - like and so on - that have easy sines and cosines, and you may not need a calculator for these. And you may be able to find two of these special angles that will combine to give (perhaps by adding, subtracting, doubling, and so on), but I can't think of any at present.
Is this what you meant?