Question

Sin(x+π/8)sin(x-π/8)​

Answer 1

1/2*[cos(x+π/8-x+π/8)-cos(x+π/8+x-π/8)]=1/2*(cosπ/4-cos2x)=

=1/2*√2/2-1/2*cos2x=√2/4-1/2*cos2x

Answer 2

To simplify the expression Sin(x+π/8) * Sin(x-π/8), we can use the trigonometric identity for the product of sines:

sin(A) * sin(B) = (1/2) * [cos(A - B) - cos(A + B)]

Let's apply this identity to the given expression:

sin(x+π/8) * sin(x-π/8) = (1/2) * [cos(x+π/8 - (x-π/8)) - cos(x+π/8 + x-π/8)] = (1/2) * [cos(x + π/8 - x + π/8) - cos(x + π/8 + x - π/8)] = (1/2) * [cos(2 * π/8) - cos(2x)] = (1/2) * [cos(π/4) - cos(2x)] = (1/2) * [(√2/2) - cos(2x)] = (√2/4) - (1/2) * cos(2x)

So the simplified expression is: (√2/4) - (1/2) * cos(2x)