How to use a simple method to derive the sum angle formula of the sine function: sin(α+β)=sinαcosβ+cosαsinβ?

• Question: Look at the 2014 Hubei Provincial College Entrance Examination Science Mathematics Questions, multiple-choice question 6:
  
  How to judge whether the first group of functions are orthogonal functions?
• Analysis: To judge whether the first group of functions are orthogonal functions, the double angle formula of the trigonometric function is used:
  
  Obviously this double angle formula is derived from the sum angle formula. The sum angle formula of the sine function in the textbook is derived from the unit circle, which is difficult to understand. It is easier to derive the area of ​​triangles in junior high school. As follows:
  
  This kind of proof is indeed simple, but it is still difficult to write quickly:
  So I recommend my own proof method, which hit the nail on the head: