Using the Sine and Cosine Addition Formulas to Prove Identities - Problem 1

I want to prove some more identities using the sine and cosine addition formulas. Let's prove what I call the add pi identities. This will give us an identity for the sine of theta plus pi, so I know theta plus pi equals blank and I want to do one for cosine as well.

Now first for the sine of theta plus pi, I need the sine of a sum identity and the sine of a sum goes sine, cosine, cosine, sine; so sine theta, cosine pi, cosine theta, sine pi. And with sine the plus stays the same remember sine same and that gives me sine of theta times cosine of pi is -1 just refer to the unit circle if you forget that and cosine of theta sine of pi is 0 and so that just gives me minus sine theta, so that's kind of interesting if you add pi to the input, you get the opposite output.

Let's see if that is true for cosine. The cosine sum identity goes cosine, cosine, sine, sine; cosine theta, cosine pi, sine theta, sine pi and with cosine remember c for change, the plus changes to minus. So it's cosine theta times -1 minus sine theta times 0. Yeah again we get minus cosine theta, so add pi and you get the opposite output, it's true for cosine too.