SOLUTION: Power reduce sin^3⁡x as a sum (or difference) of first degree trig terms

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Question 737560: Power reduce sin^3⁡x as a sum (or difference) of first degree trig terms
Answer by AnlytcPhil(1806)   (Show Source): You can put this solution on YOUR website!
The trick is to start with sin(3x) and simplify
and it will come out with an equation that you
can solve for  sin³(x).

sin(3x) = sin(2x+x) = sin(2x)cos(x)+cos(2x)sin(x) = 

2sin(x)cos(x)cos(x)+cos(2x)sin(x) =

2sin(x)cos²(x)+[1-2sin²(x)]sin(x) =

2sin(x)[1-sin²(x)]+[1-2sin²(x)]sin(x) =
  
2sin(x) - 2sin³(x) + sin(x) - 2sin³(x) =

3sin(x) - 4sin³(x)

So

sin(3x) = 3sin(x) - 4sin³(x)

4sin³(x) = 3sin(x) - sin(3x)

sin³(x) = sin(x) - sin(3x)

Edwin
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