SOLUTION: I need to prove the following identity. Basically, I need to choose one side of the problem and indicate it clearly, then work towards the solution so that both sides are eventuall

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Question 210322: I need to prove the following identity. Basically, I need to choose one side of the problem and indicate it clearly, then work towards the solution so that both sides are eventually equal.
The idea is to use relevant trig identities (Half Angle, Double Angle, Addition/Subtraction, Product-to-Sum, Sum-to-Product, etc.), but I don't know what to do!
cos 3*theta = (4 cos^3 theta) - (3 cos theta)
If you could help me towards the complete solution of this problem so I understand it, I'd appreciate it! Thanks!
Answer by tutor_paul(519) (Show Source):
You can put this solution on YOUR website!
I will rewrite this replacing theta with x to make it more clear:

I am going to work on the cos(3x) term to see if it can be made equal to the right side. The first step is kinda of unintuitive, yet it is the key to the problem. I am going to rewite cos(3x) as follows:

Notice this did not change the value of the term, since I added and subtracted the same thing (cos(x)). I did this because I want to apply the sum property, which gives me:

Now let's go double angle:

Simplify:

So there you have it... you may be discouraged by that first step and say "I would never think of that." Neither would I when I first started doing these, but if you do enough practice problems, you will see how these kind of manipulations can help you.
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Good Luck,
tutor_paul@yahoo.com