Explanation

REmember when you where in third grade and you learned what all the numbers where called in a problem? Words like Numerator and Denominator and Dividend and Divisor and such? Do you remember what the answer to a division problem was called? That's right, Quotient! So how does this work with these Identities? Well, it's simple, once you get past the big words. When we learned about   the Unit Circle, we learned that sin=Y, cos=X, tan=Y/X, cot=X/Y etc. This Identity just plugs in the names of the function, as supposed to the X/Y coordinates. Yeah, it's that simple. Take a look:

Tan(X)=Sin(X)/Cos(X)
Cot(X)=Cos(X)/Sin(X)

See? It doesn't involve anything new, they just put a new name to something we've been using for a long time.