Sum and Difference Identity Examples

Well, I basically just copied what the examples were in the green Mickelson book, but changed the numbers, but just working through the examples and my own really helped me understand them better. Here's what I came up with. By the way, I'm sure there's a way to use greek letters, but I couldn't find it so I'm substituting a for alpha and b for beta.

Example 1

Simplify: sin4xcos3x - cos4xsin3x

By using the identity sin(a - b) = sinacosb - cosbsina,

sin4xcos3x - cos4xsin3x = sin(4x - 3x)
= sin(x)

Example 2

Determine the exact value of cos(315). (That would be 315 degrees, but I don't know how to get a degrees sign on here either.)

By using the identity cos(a - b) = cosacosb + sinasinb,

cos(360 - 45) = cos360cos45 + sin360sin45
= 1(SQRT(2)/2) + 0(SQRT(2)/2)
= SQRT(2)/2

OR

By using the identity cos(a + b) = cosacosb - sinasinb,

cos(270 + 45) = cos270cos45 - sin270sin45
= 0(SQRT(2)/2) - (-1)(SQRT(2)/2)
= -(-1SQRT(2)/2)
= SQRT(2)/2

Hopefully that makes some sort of sense. Don't forget, tommorow's pi day!!