Simplifying Trig Functions Using Identities

Q:  Simplify cos4t – sin4t

Answer:

First thing we gotta notice is that this is a difference of squares.  cos4t is the same as cos2t squared.  So, we need to use what we know about factoring from algebra to factor this:

cos4t – sin4t can be factored like so (this just takes practice to see this and realize it):

(cos2t – sin2t)(cos2t + sin2t)

But, now we can use a trig identity because cos2t + sin2t = 1, so plug that in:

(cos2t – sin2t)(cos2t + sin2t)

(cos2t – sin2t)(1) = (cos2t – sin2t)

Now we gotta notice that even though it is simplified a bunch, we have another identity:

(cos2t – sin2t) = cos(2t)

And now we are as simplified as we get.