1=-1!!

The mistake is when you use $e^{i\pi} = e^{-i\pi}$ to conclude $i\pi = -i\pi$ . The exponential function $e^z$ is not a $1-1$ function on the complex plane. One cannot conclude $z_1 = z_2$ simply because $e^{z_1} = e^{z_2}.$ (Indeed, the exponential function is periodic along the imaginary axis - a central fact of complex analysis.)

The Euler equation (step 1 above) is proven with calculus by substituting Pi for x into the equation:

cos(x) + i.sin(x) = e^(i.x)

which is true for all real x.

Unfortunately, substituting -Pi gives e^(-i. Pi)=-1 which is step 4.

With no implication that i.Pi = - i.Pi

Once we have our dear friend Cos involved, we realise lots of different reals have the same cosine.

$x^y=x^z,here y=z$ ,if &only if y&z are real number, $y,z>0$ & $x$ is not equal to 1