SOLUTION: establish each identity 44) 1- 1/2sin (2 theta) = sin^3 theta + cos^3 theta over sin theta + cos theta

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Question 704213: establish each identity

44) 1- 1/2sin (2 theta) = sin^3 theta + cos^3 theta over sin theta + cos theta
Answer by AnlytcPhil(1807) About Me  (Show Source):
You can put this solution on YOUR website!
             1-+expr%281%2F2%29sin%282theta%29%22%22=%22%22%28sin%5E3%28theta%29+%2B+cos%5E3%28theta%29%29%2F%28sin%28theta%29%2Bcos%28theta%29%29

Factor the numerator on the right side as the sum of 
two cubes, using A%5E3%2BB%5E3%22%22=%22%22%28A%2BB%29%28A%5E2-AB%2BB%5E2%29

                    %22%22=%22%22

                   %22%22=%22%22

                   %22%22=%22%22sin%5E2%28theta%29-sin%28theta%29cos%28theta%29%2Bcos%5E2%28theta%29

Use the identity sin%5E2%28alpha%29%2Bcos%5E2%28alpha%29=1 on the 1st and 3rd terms:

                   %22%22=%22%221-sin%28theta%29cos%28theta%29

Use the identity sin%282alpha%29=2sin%28alpha%29cos%28alpha%29 multiplied
through by 1%2F2 or expr%281%2F2%29sin%282alpha%29=sin%28alpha%29cos%28alpha%29

                   %22%22=%22%221-sin%28theta%29cos%28theta%29

                   %22%22=%22%221-+expr%281%2F2%29sin%282theta%29

Edwin