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.
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