SOLUTION: Prove these identity (1-SinA + CosA)^2 = 2(1-SinA)(1+ CosA)

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Question 985310: Prove these identity
(1-SinA + CosA)^2 = 2(1-SinA)(1+ CosA)
Found 3 solutions by Edwin McCravy, ThePhysicsMathGuru, josgarithmetic:
Answer by Edwin McCravy(20062)   (Show Source): You can put this solution on YOUR website!
    

Square out the left side:



Replace sinē(A) by 1-cosē(A):



Remove parentheses:



Combine like terms:



Factor out 2:



Factor by grouping:  I.e., In the parentheses factor 1 out 
of the first two terms and factor cos(A) out of the last 
two terms:



Factor 1-sin(A) out of both terms:



Edwin
Answer by ThePhysicsMathGuru(3)   (Show Source): You can put this solution on YOUR website!
Rewrite the LHS as

Now Multiply Thru

Now all that is left is to collect like terms

We now end up with

Now you can either do Factoring By grouping on the previous expression or you can simply expand the RHS of the equation and the two parts are EQUAL with the 2 Factored out of the LHS
QED

Answer by josgarithmetic(39627)   (Show Source): You can put this solution on YOUR website!
Using lower case instead of A.
ID to prove:

LEFT SIDE:





-------You might or might not realize how the longer expression IS factorable.
Notice the terms 1 and -sin(a)cos(a), which may help suggest the possible factorization.

If you now take the right hand side, and keep the factor 2 separate, and multiply
the two binomials according to First, Outer, Inner, Last, you will obtain:
, essentially showing that its factorization will be
the RIGHT SIDE of the identity to be proved; and that this is identical to the
expression found when starting from the LEFT SIDE.
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