Trigonometric Expressions

TrigExpand[expr]expand trigonometric expressions out into a sum of terms
TrigFactor[expr]factor trigonometric expressions into products of terms
TrigFactorList[expr]give terms and their exponents in a list
TrigReduce[expr]reduce trigonometric expressions using multiple angles

Functions for manipulating trigonometric expressions.

This expands out a trigonometric expression:
In[1]:=
Click for copyable input
Out[1]=
This factors the expression:
In[2]:=
Click for copyable input
Out[2]=
And this reduces the expression to a form that is linear in the trigonometric functions:
In[3]:=
Click for copyable input
Out[3]=
TrigExpand works on hyperbolic as well as circular functions:
In[4]:=
Click for copyable input
Out[4]=
TrigReduce reproduces the original form again:
In[5]:=
Click for copyable input
Out[5]=
The Wolfram System automatically uses functions like Tan whenever it can:
In[6]:=
Click for copyable input
Out[6]=
With TrigFactorList, however, you can see the parts of functions like Tan:
In[7]:=
Click for copyable input
Out[7]=
TrigToExp[expr]write trigonometric functions in terms of exponentials
ExpToTrig[expr]write exponentials in terms of trigonometric functions

Converting to and from exponentials.

TrigToExp writes trigonometric functions in terms of exponentials:
In[8]:=
Click for copyable input
Out[8]=
TrigToExp also works with hyperbolic functions:
In[9]:=
Click for copyable input
Out[9]=
ExpToTrig does the reverse, getting rid of explicit complex numbers whenever possible:
In[10]:=
Click for copyable input
Out[10]=
ExpToTrig deals with hyperbolic as well as circular functions:
In[11]:=
Click for copyable input
Out[11]=
You can also use ExpToTrig on purely numerical expressions:
In[12]:=
Click for copyable input
Out[12]=