SOLUTION: Please prove: ((cos2A)/(cosA+sinA)^3) = (cosA-sinA)/(1+sin2A)

Question 1031126: Please prove: ((cos2A)/(cosA+sinA)^3) = (cosA-sinA)/(1+sin2A)
Answer by Cromlix(4381) (Show Source): You can put this solution on YOUR website!
Hi there,
((cos2A)/(cosA+sinA)^3) = (cosA-sinA)/(1+sin2A)
Take each part separately. Leave the right hand side as
the answer you are aiming for.
cos2A we are going to use cos^2A - sin^2A
Next we look at (cosA + sinA)^3
First we square (cosA + sinA)^2
= (cos^2A +2sinAcosA + sin^2A)
Consider (cos^2A +2sinAcosA + sin^2A)
(cos^2A + sin^2A = 1)
(2sinAcosA = sin2A)
So (cos^2A +2sinAcosA + sin^2A)
goes down to (1 + sin2A)
Next we still have (cosA + sinA)
to multiply into it, so we place it
next to the (1 + sin2A)
Our Left Hand side =
sin^2A - cos^2A/ (sinA + cosA)(1 + sin2A) = RHS.
Now divide (sinA + cosA) on bottom
into cos^2A - sin^2A on top
This gives you:
(cosA - sinA)/(1 + sin2A) = RHS.
Hope this helps
Sorry it is so long winded!
All the best :-)
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