Edit: Added some tex tags

This is a question on a review I am doing, my answer doesn't match up with the answer key and I am looking for some input on what I am doing wrong.

If with in quadrant IV find:

I solved for and

Using negative angle identities and the double angle identity I end up with:

Plugging in my values gives me:

Simplified to:

The answer key gives:

Not sure where I am messing up =\ any help is much appreciated.