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^@#
