Study the sine sum and difference expressions for two real angles x and y.
(i) Write the expression for sin(x+y).
(ii) Replace y by −y in your result from part (i) and simplify to obtain a formula for sin(x−y). Explain carefully how the parity (even/odd nature) of sin and cos determines the sign changes that appear during the simplification.
(iii) Using the result from part (ii), explain why sin(x−y)=0 when x=y. Give a brief geometric interpretation of this fact on the unit circle.