Question

求证:sin(a+b)cos(a-b)=sina*cosa+sinb*cosb

Answer 1

积化和差 sin(A)*cos(B) = (1/2)*[sin(A+B) + sin(A-B)]

2倍角公式 sin(2A) = 2*sinAcosA

sin(a+b)cos(a-b)

= (1/2)*{sin[(a+b)+(a-b)] + sin[(a+b)-(a-b)]}

= (1/2)*[sin(2a) + sin(2b)]

= (1/2)*[2*sin(a)*cos(a) + 2*sin(b)*cos(b)]

= sin(a)*cos(a) + sin(b)*cos(b)