Using trigonometric identities

let's do some examples simplifying trigonometric expressions so let's say that I have 1 minus sine squared theta and this whole thing times cosine cosine squared theta so how could I simplify this well the one thing that we do know this is the most fundamental trig identity this comes straight out of the unit circle is that cosine squared theta plus sine squared theta is equal to 1 and then if we subtract sine squared theta from both sides we get cosine squared theta is equal to 1 minus sine squared theta so we have two options we could either replace this 1 minus sine squared theta with the cosine squared theta or we could replace this cosine squared theta with the 1 minus sine squared theta well I'd prefer to do the former because this is a more complicated expression so if I can replace this if I can replace this with a cosine squared theta then I think I'm simplifying this so let's see this will be cosine squared theta times another cosine squared theta and so all of this is going to simplify to this is cosine theta times cosine theta times cosine theta times cosine theta well that's just going to be cosine to the fourth of theta let's do another example let's say let's say that we have let's say that we have sine squared theta sine squared theta all of that over all of that over 1 minus sine squared theta what is this going to be equal to well we already know that 1 minus sine squared theta is the same thing as cosine squared theta so it's going to be sine squared theta over over this thing is the same thing as cosine squared theta we just saw that over cosine squared theta which is going to be equal to you could view this as sine theta over cosine theta whole quantity squared but what's sine over cosine that's tangent so this is equal to tangent squared theta let's do one more example let's say that we had let's say that we have cosine squared theta plus 1 minus plus 1 minus I so let's make it this way plus 1 plus sine squared theta sine squared theta what is this going to be well you might be tempted especially with way I wrote the colors to think hey is there some identity for 1 plus sine squared theta this is really all about rearranging it to realize that G I by the unit circle definition I know what cosine squared theta plus sine squared theta is cosine squared theta plus sine squared theta for any given theta is going to be equal to 1 so this is going to be equal to 1 plus this one right over here plus this one over here which is equal to 2