I want to share this with you. The derivation of one of my favourite identites in mathematics so far:
Let f(x) = cos x = a + bx + cx^2 + dx^3 + ex^4 + fx^5 + ...
(don't mix up e in ex^4 with e = 2.7182821828...; in this case e is just any constant)
so
f(0) = cos 0 = 1 = a, thus a = 1
f'(x) = -sinx = b + 2cx + 3dx^2 + 4ex^3 + 5fx^4 + ...
f'(0) = -sin0 = 0 = b; thus b = 0
f''(x) = -cosx = (2*1)c + (3*2)d + (4*3)ex^2 + (5*4)fx^3 + ...
f''(0) = -cos0 = -1 = (2*1)c; thus c = -1/(2*1) = -1/2!
f'''(x) = sinx = (3*2*1)d + (4*3*2)ex + (5*4*3)fx^2 + ...
f'''(0) = sin0 = 0 = (3*2*1)d; thus d = 0
f''''(x) = cosx = (4*3*2*1)e + (5*4*3*2)fx + ...
f''''(0) = cos0 = 1 = (4*3*2*1)e; thus e = 1/4!
...
and so on.
Therefore substuting the constants into the original polynomial equation,
(i) cos x = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! + ...
You can check this by substitution a value in x, such as 1 (radians).
Similarly, you can work out an expansion for
(ii) sin x = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! + ...
(iii) e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + ...
Now using expansion (iii), you can expand e^(ix), where i = √(-1)
(iv) e^(ix) = 1 + ix + (ix)^2/2! + (ix)^3/3! + (ix)^4/4! + (ix)^5/5! + ...
= 1 + ix - x^2/2! - i(x)^3/3! + x^4/4! + i(x)^5/5! + ...
Separating the imaginary parts and real parts,
e^(ix) = (1 - x^2/2! + x^4/4! - x^6/6! + ...) + i(x - x^3/3! + x^5/5! - x^7/7! + ...)
= cos x + isin x
Wow!
so you get Euler's identity
And a special case of it, is when x = π
e(iπ) = cosπ + isinπ = -1 + i(0) = -1
Rearrange, then you get
an equation connecting the fundamental numbers i, π, e, 1, and 0 (zero), the fundamental operations +, ×, and exponentiation, the most important relation =, and nothing else.
Wow! Wow! Aren't you excited? (Nerd... if you actually read through all of the above and understood it.)
Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician.
It doesn't stop here though.
This opens a lot of branches in mathematics. One example is the hyperbolic function, which links circular and exponential functions - something you usually think as very separate branches of mathematics.
