Question

When you simplify:
sin(pi+x)cos((pi/2)-x)-cos(pi+x)sin(x+(3pi/2))

What's the answer cause I got -2sinxcosx, which I don't know if can simplify to -2sinx
But the answer key says -1 so. I'm not sure how to get that

Answer 1

You should know these identities:

sin(Ï€+x) = -sinx
e.g. sin 210° = sin(180°+30°) = -sin30°
cos(Ï€/2) - x) = sinx
e.g. cos(60) = cos(90-30) = sin30
cos(Ï€+x) = -cosx
e.g. cos 210° = -cos30°
sin(3Ï€/2 + x) = -cosx
e.g. sin 300° = sin(270+30) = -cos30

so ....
sin(pi+x)cos((pi/2)-x)-cos(pi+x)sin(x+(3pi/2))
= (-sinx)(sinx) - (-cosx)(-cosx)
= - sin^2 x - cos^2 x
= -1(sin^2 x + cos^2 x)
= -1(1)
= -1