Question

sin(a-b) sin(a+b) =( sina - sin b)(sina+sinb)​

Answer 1

sin(a-b)=sina*cosb-sinb*cosa

sin(a+b)=sina*cosb+sinb*cosa

sin(a-b)*sin(a+b)=(sina*cosb-sinb*cosa)(sina*cosb+sinb*cosa)=sin²a*cos²b+sina*cosb*sinb*cosa-sinb*cosa*sina*cosb-sin²b*cos²a=sin²a*cos²b-sin²b*cos²a

sin²a+cos²a=1=>cos²a=1-sin²a

sin²b+cos²b=1=>cos²b=1-sin²b

sin(a-b)*sin(a+b)=sin²a*(1-sin²b)-sin²b(1-sin²a)=sin²a-sin²a*sin²b-sin²b+sin²a*sin²b=sin²a-sin²b=(sina-sinb)(sina+sinb)

a²-b²=(a-b)(a+b)

Answer 2

$sin(a-b)*sin(a+b)=sin(2*\frac{a-b}2} )*sin(2*\frac{a+b}{2} )=\\=2*sin\frac{a-b}{2}*cos\frac{a-b}{2} *2*sin\frac{a+b}{2} *cos\frac{a+b}{2} \\=2*cos\frac{a+b}{2}*sin\frac{a-b}{2}*2*sin\frac{a+b}{2}*cos\frac{a-b}{2}\\=(sina-sinb)(sina+sinb)\\Bafta!$