Simplify the cos of 𝐴 minus 𝐵 minus the cos of 𝐴 plus 𝐵.
There are two trig identities that we know that we could use: the cos of 𝑥 minus 𝑦 is equal to the cos of 𝑥 times the cos of 𝑦 plus the sin of 𝑥 times the sin of 𝑦 and the cos of 𝑥 plus 𝑦 is equal to the cos of 𝑥 times the cos of 𝑦 minus the sin of 𝑥 times the sin of 𝑦.
So essentially, all that we have to do is replace 𝑥 with 𝐴 and 𝑦 with 𝐵. So we will wanna take each of these pieces and subtract them, since that’s what we have in our original expression. So let’s go ahead and replace 𝑥 with 𝐴, 𝑦 with 𝐵, and then we subtract.
So now that we’ve substituted in, we have to be careful with this minus sign. We need to distribute it to both of the terms inside the brackets. So after we’ve distributed the negative sign, now we can combine like terms: cos 𝐴 cos 𝐵 are exactly the same, except one is a minus, so they will cancel.
And then we have two identical sin 𝐴 sin 𝐵s, so we have two of them, so we can add them together. So since we have two terms that are exactly the same, we have two of them, so we can write two sin 𝐴 sin 𝐵, and this would be our final answer.