Establish the identity sin^2 A - sin^2 B

Question

Establish the identity

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

Answer 1

Hi Katelyn!

Addition & Subtraction identities

Sin(A-B) = SinACosB - CosASinB

Sin(A+B) = SinACosB +CosASinB

Sin(A-B) x Sin(A+B) =

(SinACosB-CosASinB)(SinACosB+CosASinB)

Use FOIL

Sin²ACos²B+SinASinBCosACosB -

SinASinBCosACosB-Cos²ASin²B

Middle terms drop out

(1) Sin²ACos²B - Cos²ASin²B

Sin²A + Cos²A = 1 pythagorean identities

Sin²B + Cos²B = 1

Cos²B=(1 - Sin²B), Cos²A=(1 - Sin²A)

Substitute into equation (1)

Sin²A(1-Sin²B) - (1-Sin²A)Sin²B

Multiply through parentheses...

Sin²A - Sin²ASin²B - (Sin²B - Sin²ASin²B)

Sin²A - Sin²ASin²B - Sin²B + Sin²ASin²B

The longer two terms cancel out and you are

left with what you are trying to prove.

Sin²A - Sin²B

Answer 2

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

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

Use algebraic identity (x-y)(x+y) = x²-y² where x = sin(A)cos(B) and y = sin(B)cos(A)

sin²(A)cos²(B)-sin²(B)cos²(A)

sin²(A)(1-sin²(B)) - sin²(B)(1-sin²(A))

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

sin²(A) - sin²(B)