Use an Addition or Subtraction Formula to complete the follo

Use an Addition or Subtraction Formula to complete the following two steps: write the expression as a trigonometric function of one number, and then find its exact value: sin 7 pi/12 cos pi/12 - cos 7 pi/12 sin pi/12 cos 5pi/18 cos pi/9 + sin 5 pi/18 sin pi/9 sin 22^cos 23^+ cos 22 sin 23 tan 104^- tan 44/1 + tan 104^tan 44

Solution

sin(7pi/12)cos(pi/12)-cos(7pi/12)sin(pi/12)

This is of the form

sin A cos B - cos A sinB which is equal to sin(A-B)

so we get

sin(7pi/12 - pi/12)= sin(pi/2)=1

cos 5pi/18 cos pi/9 + sin pi/18 sin pi/9

This is of the form cos A cos B + sinA sin B which is equal to cos (A-B)

so we get cos(5pi/18-pi/9)=cos(3pi/18)=cos pi/6=sqrt3/2

sin 22 cos 23 + cos 22 sin 23= sin(22 +23)=sin 45= sqrt2/2