Verify the identity sin^2b+sin(a+b)*sin(a-b)=sin^2a.

We'll re-write the product sin(a+b)*sin(a-b), using the
formulas for the sum and difference of the angles a and
b.

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)

We notice that the
product is the difference of squares:

(sina*cosb +
sinb*cosa)(sina*cosb - sinb*cosa) = (sina*cosb)^2 -
(sinb*cosa)^2

We'll re-write the identity, substituting the
product:

(sinb)^2 + (sina*cosb)^2 - (sinb*cosa)^2 =
(sina)^2

We'll subtract (sina*cosb)^2 both
sides:

(sinb)^2- (sinb*cosa)^2 = (sina)^2 -
(sina*cosb)^2

We'll factorize by sin b to the left side and
we'll factorize by sin a, to the right side:

sin b(1 -
(cosa)^2) = sin a(1 - (cosb)^2)

sin b(sin a)^2 = sin a(sin
b)^2

We'll divide by sin b and sin a both
sides:

sina=sinb