SOLUTION: Find the value of the trigonometric expression: sin 50degrees cos 170degrees-cos 50degrees sin 170degrees also: cos 150degress cos 120degrees-sin 150degrees sin 120degre

You have to know these identities backward and forward, and
notice carefully the +'s and -'s:

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
 
sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

sin(-A) = -sin(A)

cos(-A) = cos(A)

Then when you have learned those, you will be able to
look at

      sin(50°)cos(170°) - cos(50°)sin(170°)

and recognize that it looks like the right side of

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

with 50° substituted for A and 170° substituted for B, so
make those substitutions in both sides of

      sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

and get

sin(50° - 170°) = sin(50°)cos(170°) - cos(50°)sin(170°)

Simplify the left side

sin(-120°)

and then if you know 

sin(-A) = -sin(A)

-sin(120°) 

and if you know the special angles, you know that
equals:
    _
   √3
- ————
    2

---------------------------------

Also when you have learned those identities, you will be 
able to look at

      cos(150°)cos(120°) - sin(150°)sin(120°)

and recognize that it looks like the right side of

     cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

with 150° substituted for A and 120° substituted for B, so
make those substitutions in both sides of

     cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

and get

cos(150° + 120°) = cos(150°)cos(120°) - cos(150°)sin(120°)

Simplify the left side

cos(270°)

and if you know the special angles, you know that
equals:

 0

Edwin