Derive most trigonometric identities from the addition formulas

[hypermonkey2]

In the same way that it is possible to derive most trigonometric identities from the addition formulas, what is the way that the difference of sines and cosines formulas were derived, such as

$$\sin{a}-\sin{b}=2\cos{\frac{a+b}{2}}\sin{\frac{a-b}{2}}$$

thanks, I am trying to avoid as much memorization as possible, if anyones wondering.

 

[vaishakh]

Take a = c + d and b = c - d for arithmetic mean c. Now substitute sinc+d and subtract sinc-d form it. the sind*cosc have opposite signs in bot the formula. so the answer is 2sind*cosc which when substituted you get above result.
similarly you can do for sina + sinb and addition as well as subtraction for cos as well as tan.

 

[hypermonkey2]

interesting! so the trick is to suppose that a and b are at equal distance from an intermediate value, correct? Very nice solution, thanks.