Use Euler's Identity to prove?

Question

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.

Answer 1

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.