SOLUTION: please prove: {{{(sin(x)+sin(3x))/(cos(x)+cos(3x))=tan(2x)}}}

Work with only the left side:

Write  as  and  as 



Use the identities

and 

to rewrite the numerator and
use the identities 

and


to rewrite the denominator 



Remove the parentheses



Cancel:






The two terms on top are like terms, and
the two terms on the bottom are also like terms



Cancel the 2'a and the cosines:





Use identity 



Edwin