Here is an example of using a sum identity:
Find sin15^@.
If we can find (think of) two angles A and B whose sum or whose difference is 15, and whose sine and cosine we know.
sin(A-B)=sinAcosB-cosAsinB
We might notice that 75-60=15
so sin15^@=sin(75^@-60^@)=sin75^@cos60^@-cos75^@sin60^@
BUT we don't know sine and cosine of 75^@. So this won't get us the answer. (I included it because when solving problems we DO sometimes think of approaches that won't work. And that's OK.)
45-30=15 and I do know the trig functions for 45^@ and 30^@
sin15^@=sin(45^@-30^@)=sin45^@cos30^@-cos45^@sin30^@
=(sqrt2/2)(sqrt3/2)-(sqrt2/2)(1/2)
=(sqrt6 - sqrt 2)/4
There are other way of writing the answer.
Note 1
We could use the same two angles and the identity for cos(A-B) to find cos 15^@
Note 2
Instead of 45-30=15 we could have used 60-45=15
Note 3
Now that we have sin 15^@ we could use 60+15=75 and sin(A+B) to find sin75^@. Although if the question had been to find sin75^@, I'd probably use 30^@ and 45^@#
