**Q: Simplify cos**^{4}t – sin^{4}t

A:

First thing we gotta notice is that this is a difference of squares. **cos**^{4}t is the same as (**cos**^{2}t)^{2}. So, we need to use what we know about factoring from algebra to factor this:

**cos**^{4}t – sin^{4}t can be factored using the difference of squares formula/concept (this just takes practice to see this and realize it):

**(cos**^{2}t – sin^{2}t)(**cos**^{2}t + sin^{2}t)

But, now we can use a trig identity because **cos**^{2}t + sin^{2}t = 1, so plug that in:

**(cos**^{2}t – sin^{2}t)(**cos**^{2}t + sin^{2}t)

**(cos**^{2}t – sin^{2}t)(**1) = ****(cos**^{2}t – sin^{2}t)

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

**(cos**^{2}t – sin^{2}t) = cos(2t)

And now we are as simplified as we get.

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