Euler's Identity
The greatest identity on Earth, relating the base of the natural logarithm e, with the elementary trigo functions. substituting pi into the identity immediately yields;
An equation relating the five most important numbers in mathematics. Nothing else needs to be said. I will attempt to give a single proof of this today. Which is not really totally formal, but works nonetheless.Proof.
Consider the function
f(x)=
Differentiating we will have,
f '(x)=
f '(x) = 0
Take a moment to think about what this means. This tells us that the gradient of f(x) at Every point of its graph is equal to zero. Thus is we draw the graph it must look like a straight horizontal line. Which means that it must be a constant function.
For every value of x, fx is a constant. Thus we can find out what this value is by substituting any value for x. Lets try zero.
f(x)=constant=f(0)= 
which reduces to one.
Thus we have shown, f(x)=1 for all x,
is equal to one, for all x
Yupp. Too chio.
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