SOLUTION: prove that: 2 sin^2(45-A)=1-sin2A

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Question 1140881: prove that:
2 sin^2(45-A)=1-sin2A
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
2sin%5E2%2845-A%29=1-sin%282A+%29

Manipulating left side:

2sin%5E2%2845-A%29

since sin%2845-A%29=+%28sqrt%282%29%2F2%29cos%28A%29-%28sqrt%282%29%2F2%29sin%28A%29, we have

=2%28%28sqrt%282%29%2F2%29cos%28A%29-%28sqrt%282%29%2F2%29sin%28A%29%29%5E2...simplify, factor out %281%2F2%29%5E2=1%2F4
=2%281%2F4%29%28sqrt%282%29cos%28A%29-sqrt%282%29sin%28A%29%29%5E2...simplify, factor out %28sqrt%282%29%29%5E2=2

=cross%282%29%281%2Fcross%284%292%29%282%28cos%28A%29-sin%28A%29%29%5E2%29

=%281%2Fcross%282%29%29%28cross%282%29%28cos%28A%29-sin%28A%29%29%5E2%29

=%28cos%28A%29-sin%28A%29%29%5E2.......expand

=cos%5E2%28A%29-2sin%28A%29cos%28A%29%2Bsin%5E2%28A%29.........use identity sin%5E2%28A%29%29%2Bcos%5E2%28A%29=1
=1-2sin%28A%29cos%28A%29 .......use identity 2sin%28A%29cos%28A%29+=sin%282A+%29

=1-sin%282A+%29