SOLUTION: Verify the following identity: Sin^4x + 2sin^2xcos^2x + cos^4x = 1

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Question 623872: Verify the following identity:
Sin^4x + 2sin^2xcos^2x + cos^4x = 1
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sin%5E4%28x%29+%2B+2sin%5E2%28x%29cos%5E2%28x%29+%2B+cos%5E4%28x%29+=+1
I'm sure there are other ways to do this but probably the easiest is to recognize that the left side is a perfect square trinomial. It fits the pattern of:
a%5E2%2B2ab%2Bb%5E2+=+%28a%2Bb%29%5E2
with the "a" being sin%5E2%28x%29 and the "b" being cos%5E2%28x%29. Once we recognize this, the rest is easy. Factoring the left side according to the pattern we get"
%28sin%5E2%28x%29+%2B+cox%5E2%28x%29%29%5E2+=+1
We should know well that sin%5E2%28x%29+%2B+cox%5E2%28x%29 = 1, so this becomes:
1%5E2+=+1
which simplifies to:
1+=+1
And we're finished!!