Prove:
cos4x = 8cos^4x - 8cos^2x + 1
My Attempt:
RS:
= 4cos^2x (2cos^2x - 1) + 1
= 4 cos^2x (cos2x) + 1
LS:
= cos2(2x)
= 2cos^2(2x) - 1
= (cos^2(2)) - cos^2(2x)) - 1
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Prove:
8cos^4x = cos4x + 4cos2x + 3
My Attempt:
RS:
= cos2(2x) + 4cos(2x) + 3
= 2cos^2(2x) - 1 + 2(4)cos^2(2x) - 1 + 3
When proving these, you should only change ONE side of the equation. Pick one side, and try to get it to match the other.
For the first one, it's important to note the double-angle identity for cosines. There are three versions, but since both sides contain cosines, we're going to use the version that includes only cosines.
cos(2x) = 2cos^2(x) - 1
cos(4x) = 8cos^4(x) - 8cos^2(x) + 1
Since the RS seems like an expansion of the LS, we will work SOLELY with the LS to get it to match the RS.
cos(4x)
cos(2*2x) Double-angle identity.
2cos^2(2x) - 1 Double-angle again.
2(cos2x * cos2x) - 1 Double-angle again.
2((2cos^2(x) - 1)(2cos^2(x) - 1)) - 1
Now, distribute the 2 into the first parentheses.
(4cos^2(x) - 2)(2cos^2(x) - 1) - 1
Foil.
8cos^4(x) - 4cos^2(x) - 4cos^2(x) + 2 - 1
Simplify.
8cos^4(x) - 8cos^2(x) + 1
That matches the RS, so we're done.
Try a similar process with your other proof. These do take a bit of practice. You may need to practice with more than the amount of homework that your teacher has assigned. It took me an extra 10 hours of additional practice in a week when we first learned these. Once you do enough, you'll see the patterns. It's crucial to memorize the identities, though.
Why is cos4x, cos(2*2x) instead of cos2(2x)? or do both mean the same thing?
cos2(2x) should be cos(2(2x)). That 2 can't be floating around outside the cosine function. In any case, cos(2(2x)) is the same as cos(2*2x).
For the following lines,
2cos^2(2x) - 1 Double-angle again.
2(cos2x * cos2x) - 1 Double-angle again.
I don't get how you got the second line from the first line...
I was looking at another of your questions, and in Npgm's proof, he/she modified both sides of the equation. You absolutely cannot do this. You must change only one side of the equation. My teacher called this "math etiquette," but it is etiquette that must be followed. (I just wanted to stress this again.)
Okay? But, I still don't get what you did in:
2cos^2(2x) - 1 Double-angle again.
2(cos2x * cos2x) - 1 Double-angle again.
I'll try to explain the two lines better...
2cos^2(2x) - 1
This is like 2(cos^2(2x)) - 1.
In this line, we're changing only the cos^2(2x). When the squared is written on the "cos" part, that means the entire function is squared. Therefore, cos^2(2x) is the same as (cos(2x))^2.
When something is squared, you multiply it by itself. For example, 4^2 = 4*4, and (n+1)^2 = (n+1)(n+1).
Therefore, we are saying that (cos(2x))^2 = (cos(2x))(cos(2x)) = cos2x * cos2x. The rest of the line remained unchanged.