Verified answer

Use the compound trig form, and we have...

LHS

= (cos(x)cos(π/4) + sin(x)sin(π/4))cos(π/4) - (cos(x)cos(π/4) - sin(x)sin(π/4))sin(π/4)

= cos(π/4)(cos(x)cos(π/4) + sin(x)sin(π/4) - cos(x)cos(π/4) + sin(x)sin(π/4)) [Noting that cos(π/4) = sin(π/4)]

= cos(π/4) * 2sin(x)sin(π/4)

Note that sin(2x) = 2cos(x)sin(x). Then, letting x = π/4, we have...

2cos(π/4)sin(π/4) * sin(x)

= sin(2 * π/4)sin(x)

= sin(π/2)sin(x)

= sin(x)

= RHS

Good luck!