Question

2sin(45 + x) cos(45 + y)= cos(x + y) + sin(x - y)

Answer 1

Left Hand Side:
=2(sin45*cosX + cos45*sinX)(cos45*cosY - sin45*sinY)
=2*(cosX*(1/root2) + sinX*(1/root2))*(cosY*(1/root2) - sinY*(1/root2))
=2*(cosX + sinX)*(cosY-sinY)*(1/root2)*(1/root2)
=2*(cosX + sinX)*(cosY-sinY)*(1/2)
=(cosX + sinX)*(cosY-sinY)
=cosX*(cosY - sinY) + sinX*(cosy - sinY) 
=cosX*cosY - cosX*sinY + sinX*cosY - sinX*sinY
=cosX*cosY - sinX*sinY + sinX*cosY - cosX*sinY
=cos(X+Y) + sin(X-Y)