Two ways to find the cosine of 180 degree angle I found a question how to find the value of cos 180, then we all know that its answer is equal to cos 0, which give us 1 as answer. I myself think that the idea of cos 180 is equal to 1 is :

cos 180 = cos(180 - 0) 
cos 180 = -cos 0        "which is cos(180-a) =- cos a"
cos 180 =- 1.          

cos 180 = cos(270-90) 
cos 180 = -sin 90      cos(270-a) = -sin a, 
cos 180 = -1
In the second line you are using $\cos(A-B)=\cos A\cos B+\sin A\sin B$ but the problem is that you are implying $\cos180=1$ ($\sin180=0$ so that part cancels) to end up with the result $\cos180=1$, which makes your proof "correct", but obviously is incorrect.

There are lots more examples where an incorrect assumption while using a correct identity can make you "prove" that assumption to be true.
cos180 = cos(90+90)
Using cos(A+B) = cosAcosB-sinAsinB

cos(90+90) = cos90cos90-sin90sin90

       = 0*0-1*1

       =-1


therefore, cos180 = -1