Derivatives: A Real World Application

So, you know how people are always like ____+ _____ = 4ever, or something like that?  Well, I did the same thing with me and M.  M + T = 4 ever.  Then I realized " wait a second -- that looks like an equation! Hmmm....."  So, I started messing with said "equation".  First, I decided that it could be a whole lot more interesting, at least from a mathematics perspective, if it was multiplied. , So the equation was now...

M x T= 4Ever

This, of course, led to some interesting things, such as:

M= 4Ever/T

T=4Ever/M

Thus,

M x T = 4Ever, but, T= 4Ever/M

Thus,

M x(4Ever/M)=4Ever

Cancel out the M's, and

4Ever=4Ever

Then I  thought "Hmmmm....what if I tried to differentiate this equation.  Well.....with the equation as it is, it didn't work too well......I started out with each of the parts being a constant, which of course left the derivative as 0.  Then, I decided to assign each part of the equation its own "function" (not exactly what it was, but I can't think of the proper term.).   Now, I'm a math geek, and so, of course, I've heard the stupid pick up line about someone being sin^2 X, and the other person being cos^2 x, and together they're one.   So, I went back to the original M + T= 4Ever, and assigned each thing a value:

M=sin^2 x

T=cos^2 x

4Ever= 1

Then, we have:

sin^2x + cox^2x =1

Take the derivative, and it becomes.....

2sinx -2cosx = 0

Add 2 cos x to both sides, and....

2sinx = 2cos x

Cancel the 2

sin x = cos x

Awww......so, apparently, if M and I were both these functions, and we took the derivatives, we'd be equal... lol.  :) From a math perspective, at first glance, this looks to be a false statement.  But wait!! what about 45 degrees (pi/4 for those who prefer radians)?  isn't the cosine value the same as the sine value there?  and what about the angle in the third quadrant?  215 degrees? (5pi/4).  And then of course there's infinitely many coterminal angles.....

Thus, using derivatives, trigonometry, I have proved that M' and T' are equal.  lol.  Well, that's My Crazy Life!