sin(pi+x) cos(pi/2-x) - cos(pi+x) sin(x+3pi/2)

Answer 1

draw it
sin (pi+x) = -sin x
cos (pi/2 - x) = sin x
cos(pi+x) = -cos x
sin(1.5pi + x ) = -cos x
so
-sin x* sin x - (-cos x)* (- cos x)
= - (sin^2 x + cos^2 x)
= -1

Answer 2

Use trigonometric identities:

sin ( π + x ) = - sin x

cos ( π / 2 - x ) = sin x

cos ( π + x ) = - cos x

sin ( x + 3 π / 2 ) = - cos x

So

sin ( π + x ) ∙ cos ( π / 2 - x ) - cos ( π + x ) ∙ sin ( x + 3 π / 2 ) =

- sin x ∙ sin x - ( - cos x ) ∙ ( - cos x ) =

- sin² x - cos² x = - ( sin² x + cos² x ) = - 1