sin(A+B) sin(A-B)=sin^2(A)-sin^2(B)

Question

Help please

Answer 1

It seems this is an identity proof problem.  We want the left side equal to the right side.  Simply the left side using the addition and subtraction angle identity.

 

 

 

sin(A + B) sin(A - B) =

 

(sin(A)cos(B) + cos(A)sin(B))(sin(A)cos(B) - cos(A)sin(B)) =

 

sin²(A)cos²(B) - cos²(A)sin²(B)

 

 

Since the right side is in terms of sine, we want the left side to be in terms of sine.  We can use the identity

sin²θ + cos²θ = 1.

 

sin²(A)[1 - sin²(B)] - [1 - sin²(A)]sin²(B) =

 

sin²(A) - sin²(A)sin²(B) - sin²(B) + sin²(A)sin²(B)

 

 

The terms coded in orange cancel each other out.  We are left with

 

sin²(A) - sin²(B)

 

This simplify version of the left side matches the right side.  This checks out.