THE TRICK  
To learn the derivatives of the 6 trig functions you actually only have to learn 3 of them. The three to learn are sine, tangent, and secant. 



The other three functions all start with "co": Cosine, Cotangent, and Cosecant. Think of the functions as having partners: 



To find the derivative of the "co" functions, start with the derivatives of sine, tangent and secant, change each function in the derivative to it’s cofunction partner and put a minus sign in front of the derivative. 

So starting with one of the three we need to know: If F(x) = secant X then F'(x) = secant X tangent X


So now to get the derivative of the cosecant: If F(X) = cosecant X then F'(X) = cosecant X cotangent X 

So all 6 trig function derivatives look like this: 


Thanks to Bruce Kirkpatrick for this.