what is the difference from these two equations?

Question

i have e^ix = cos(x) + i sin(x) and e^i(pi) + 1 = 0(euler's identity)

Answer 1

well e*i(pi) is the first equation with a value substituted for x, in this case pi. rewrite your second equation as e^i(pi) = -1
now go back to your other equation
e^i(pi) = cos(pi) + i sin(pi)
cos(pi)= -1 and sin(pi) = 0
so e^i(pi) = -1 + i * 0 = -1

Answer 2

1 = sin^2 + cos^2
also you got a pi in one and not the other. One is a formula, the other is not.

Answer 3

e^ix = cos(x) + i sin(x) is the general form of the equation for all values of x

substitute x=pi

e^i(pi) = cos(pi) + isin(pi)

cos(pi) = -1
sin(pi) = 0

e^i(pi) = -1+0=-1
e^i(pi) + 1 = 0
Euler's identity is a specific form of the equation when x=pi

Answer 4

One is a general case and one is a special case when x = pi.

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

When x = pi,

e^(i * pi) = cos(pi) + i sin(pi)
e^(i * pi) = -1 + i (0)
e^(i * pi) = -1
e^(i * pi) + 1 = 0

Answer 5

e^(ix) = cos x + i sin x
e^(iπ) + 1 = cos(π) + i sin(π) + 1 = - 1 + 0 + 1 = 0

Answer 6

In the second you have substituted the value of x otherwise both are one and the same.