Use Euler's Identity to prove?
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Use Euler's identity to prove the pythagorean identity, cos^2(x) +sin^2(x) =1 I have (cos(x))(cos(x)) = 1-sin^2(x) Can someone inform me as to the next step? Help would be appreciated.
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Answer:
e^(ix) = cos(x) + i*sin(x). e^(-ix) = cos(-x) + i*sin(-x) = cos(x) - i*sin(x). Multiply the two together: LHS = e^(ix)*e^(-ix) = e^(ix - ix) = e^0 = 1. RHS = (cos(x) + i*sin(x))(cos(x) - i*sin(x)) = cos^2(x) - i*sin(x)cos(x) + i*sin(x)cos(x) - i^2 sin^2(x) = cos^2(x) + 0 - (-1)sin^2(x) = cos^2(x) + sin^2(x). Since LHS = RHS, we have cos^2(x) + sin^2(x) = 1.
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