Question

Using the fact that sin ( A + B ) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

Answer 1

Given fact is sin( A+B )=sinAcosB+cosAsinB.

Differentiate both sides with respect to x as,

d dx [ sin( A+B ) ]= d dx ( sinAcosB )+ d dx ( cosAsinB ) cos( A+B ) d dx ( A+B )=cosB d dx ( sinA )+sinA d dx ( cosB )+sinB d dx ( cosA ) +cosA d dx ( sinB ) cos( A+B )[ dA dx + dB dx ]=cosBcosA dA dx +sinA( −sinB ) dB dx +sinB( −sinA ) dA dx +cosAcosB dB dx cos( A+B )[ dA dx + dB dx ]=( cosAcosB−sinAsinB )×[ dA dx + dB dx ]

Further simplify,

cos( A+B )=( cosAcosB−sinAsinB )

Hence, the formula for cosine is proved.