Eleven Standard >> Product formula of compound angles

Product formula of compound angles

\(\cos(A+B)\cos(A-B)\)
               =\((cosAcosB-sinAsinB)(cosAcosB+sinAsinB)\)
               =\(\cos^{2}A\cos^{2}B-\sin^{2}A\sin^{2}B\)
               =\((1-\sin^{2}A)(1-\sin^{2}B)-\sin^{2}A\sin^{2}A\)
               =\(1-\sin^{2}A-\sin^{2}B+\sin^{2}A\sin^{2}B-\sin^{2}A\sin^{2}B\)
               =\(\cos^{2}A-\sin^{2}B\)

  OR  \(\cos^{2}B-\sin^{2}A\)

\(\sin(A+B)\sin(A-B)\)
            =\((sinAcosB+cosAsinB)(sinAcosB-cosAsinB)\)
            =\((sinAcosB)^2-(cosAsinB)^2\)
            =\(\sin^{2}A\cos^{2}B-\cos^{2}A\sin^{2}B\)
            =\(\sin^{2}A(1-\sin^{2}B)-(1-\sin^{2}A)\sin^{2}B\)
            =\(\sin^{2}A-\sin^{2}A\sin^{2}B-\sin^{2}B+\sin^{2}A\sin^{2}B\)
            =\(\sin^{2}A-\sin^{2}B\)
            =\((1-\cos^{2}A)-(1-\cos^{2}B)\)
            =\(\cos^{2}B-\cos^{2}A\)

Thus \(\sin(A+B)\sin(A-B)\)
             =\(\sin^{2}A-\sin^{2}B\)
             or  \(\cos^{2}B-\cos^{2}A\)