Using the identity cos(A+B)= cosAcosB-sinAsinB, prove that cos2A=1-2sin^2A.

Use cos(A+B)=cosAcosB-sinAsinB and let A=B so cos(A+A)=cosAcosA-sinAsinA this means cos(2A)=cos²A-sin²A and since cos²A+sin²A=1, cos²A=1-sin²A. Therefore, by subbing cos²A=1-sin²A into cos(2A)=cos²A-sin²A, we get cos(2A)=1-sin²A-sin²A=1-2sin²A.