One way it's useful is in the application of phasors in physics. Phasors require you to add up vectors of two different angles, so the cosine angle addition formula can be used if you already know the cosine and sine of the original vector angles.
The identity property means that whatever you put in is whatever comes out. Notice that if we multiply 1 with any number, we get that number back. So multiplying by 1 is the identity property of multiplication.
