Question

Докажите тождество:
sin(α+β)∗sin(α-β)=sin^2α-cos^2β
* -- умножить
^2 -- во второй степени

Answer 1

$Sin( alpha + beta )*Sin( alpha - beta )= frac{Cos( alpha + beta - alpha + beta )-Cos( alpha + beta + alpha - beta )}{2}=$$frac{Cos2 beta -Cos2 alpha }{2} = frac{1-2Sin ^{2} beta -1+2Cos ^{2} alpha }{2} = frac{2(Cos ^{2} alpha -Sin^{2} beta ) }{2} =Cos ^{2} alpha -Sin ^{2} beta$=$1-Sin ^{2} alpha -1+Cos^{2} beta = Cos ^{2} beta -Sin ^{2} alpha$