SOLUTION: verify the identity algebraically sin3x=sinx(3-4sin^2x)

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Question 596433: verify the identity algebraically
sin3x=sinx(3-4sin^2x)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sin%283x%29+=+sin%28x%29%283-4sin%5E2%28x%29%29
There are a number of things to consider when trying to figure out these identities:
  • Match arguments -- Use argument-changing Trig properties (2x, 1/2x, A+B, A-B) to change arguments on the left to match those on the right
  • Match the number of terms -- Use algebra and/or Trig properties to get the same number of terms on the left side as there are on the right.
  • Match functions -- Use Trig properties to change functions on the left to match those on the right.
  • If none of the above look like they are going to work, try using Trig properties to change sec, csc, tan or cot on the left into sin and/or cos.
Unfortunately, I cannot give a recipe of what to do. You just have to know your properties and algebra well enough to see which of the above looks promising. (If nothing looks promising, change everything into sin's and/or cos's and look again.)

One of the first things I notice is that the argument on the left is 3x and there are no 3x's on the right side. So we need to change the arguments. This is where we will start. There is no sin(3x) formula. (Although, if we prove this identity, we could use this equation.) What we can do, is rewrite 3x as x + 2x and then use the sin(A+B) formula:
sin%28x+%2B+2x%29+=+sin%28x%29%283-4sin%5E2%28x%29%29
sin%28x%29cos%282x%29+%2B+cos%28x%29sin%282x%29+=+sin%28x%29%283-4sin%5E2%28x%29%29
We still have some arguments to change. We don't want the 2x's any more than we wanted the 3x. But 2x is closer to x than 3x so we have made progress. For sin(2x) we have just one formula. But for cos(2x) there are three choices:
  • cos%5E2%28x%29-sin%5E2%28x%29
  • 2cos%5E2%28x%29-1
  • 1-2sin%5E2%28x%29
Since the right side has only sin's, I'm going to choose the the last one (with just sin):

Simplifying we get:


After all that we now have the arguments we want: x. Next we will match the functions. We only want sin. The only non-sin is cos%5E2%28x%29 and we can use a Pythagorean property to turn it into an expression of sin(x):

Simplifying we get:

Combining like terms we get:
3sin%28x%29+-+4sin%5E3%28x%29+=+sin%28x%29%283-4sin%5E2%28x%29%29

Now that we have the arguments and functions matched, we just have to find a way to make the whole left side match the right side. We can see that sin(x) is a factor on the right side. So we want to have sin(x) be a factor on the left side, too. Fortunately sin(x) is a factor of the left side:
sin%28x%29%283+-+4sin%5E2%28x%29%29+=+sin%28x%29%283-4sin%5E2%28x%29%29
And we're done!