Derivative Rules for y=cos(x) and y=tan(x)
Key Questions

The derivative of
#tanx# is#sec^2x# .To see why, you'll need to know a few results. First, you need to know that the derivative of
#sinx# is#cosx# . Here's a proof of that result from first principles:Once you know this, it also implies that the derivative of
#cosx# is#sinx# (which you'll also need later). You need to know one more thing, which is the Quotient Rule for differentiation:Once all those pieces are in place, the differentiation goes as follows:
#d/dx tanx#
#=d/dx sinx/cosx# #=(cosx . cosxsinx.(sinx))/(cos^2x)# (using Quotient Rule)#=(cos^2x+sin^2x)/(cos^2x)# #=1/(cos^2x)# (using the Pythagorean Identity)#=sec^2x# 
Using the definition of a derivative:
#dy/dx = lim_(h>0) (f(x+h)f(x))/h# , where#h = deltax# We substitute in our function to get:
#lim_(h>0) (cos(x+h)cos(x))/h# Using the Trig identity:
#cos(a+b) = cosacosb  sinasinb# ,we get:
#lim_(h>0) ((cosxcos h  sinxsin h)cosx)/h# Factoring out the
#cosx# term, we get:#lim_(h>0) (cosx(cos h1)  sinxsin h)/h# This can be split into 2 fractions:
#lim_(h>0) (cosx(cos h1))/h  (sinxsin h)/h# Now comes the more difficult part: recognizing known formulas.
The 2 which will be useful here are:
#lim_(x>0) sinx/x = 1# , and#lim_(x>0) (cosx1)/x = 0# Since those identities rely on the variable inside the functions being the same as the one used in the
#lim# portion, we can only use these identities on terms using#h# , since that's what our#lim# uses. To work these into our equation, we first need to split our function up a bit more:#lim_(h>0) (cosx(cos h1))/h  (sinxsin h)/h# becomes:
#lim_(h>0)cosx((cos h1)/h)  sinx((sin h)/h)# Using the previously recognized formulas, we now have:
#lim_(h>0) cosx(0)  sinx(1)# which equals:
#lim_(h>0) (sinx)# Since there are no more
#h# variables, we can just drop the#lim_(h>0)# , giving us a final answer of:#sinx# . 
#dy/dx=(cos x)'=sin x# . See derivatives of trig functions for details.
Questions
Differentiating Trigonometric Functions

Limits Involving Trigonometric Functions

Intuitive Approach to the derivative of y=sin(x)

Derivative Rules for y=cos(x) and y=tan(x)

Differentiating sin(x) from First Principles

Special Limits Involving sin(x), x, and tan(x)

Graphical Relationship Between sin(x), x, and tan(x), using Radian Measure

Derivatives of y=sec(x), y=cot(x), y= csc(x)

Differentiating Inverse Trigonometric Functions