1.  In class we used figure 1 to prove  the formula sin(x+y)=sin(x)cos(y)+cos(x)sin(y), assuming  that
x,  y and x+y are acute.

      FIGURE 1
 

Use figure 1  to  prove that

cos(x+y)=cos(x)cos(y)-sin(x)sin(y).
 

2. Carry out the following construction with GSP.   You will use it in problems 3 and 4 to prove the subtraction  formulas.

Construct a quadrilateral ABCD,  such that
angle  ABC and angle CDA are right angles and
angle DAB is acute.

(Construction tip:   Make a circle with diameter AC.
Then use what we have learned about triangles inscribed in a semicircle.)

Draw the diagonal AC.

Drop a perpendicular from B to AD,  and let E denote the point where the perpendicular
hits AD.

Drop a perpendicular from C to BE,  and let F denote the point where the perpendicular
hits BE.

Clean up your picture by hiding all extraneous circles, rays,  etc..   Label your figure  FIGURE 2.

3.  Referring to figure 2,  let x denote angle DAB, and let y denote angle CAB.   Thus angle CAD is x-y.

Use figure 2 to prove

sin(x-y)=sin(x)cos(y)-cos(x)sin(y)

4. Use figure 2 to prove

cos(x-y)=cos(x)cos(y)+sin(x)sin(y).