Question 235431
{{{(sin(x)+sin(3x))/(cos(x)+cos(3x))=tan(2x)}}}
<pre><font size = 4 color = "indigo"><b>
Work with only the left side:

Write {{{x}}} as {{{2x-x}}} and {{{3x}}} as {{{2x+x}}}

{{{(sin(2x-x)+sin(2x+x))/(cos(2x-x)+cos(2x+x))=tan(2x)}}}

Use the identities
{{{sin(alpha-beta)= sin(alpha)cos(beta)-cos(alpha)sin(beta)}}}
and 
{{{sin(alpha+beta)= sin(alpha)cos(beta)+cos(alpha)sin(beta)}}}
to rewrite the numerator and
use the identities 
{{{cos(alpha-beta)= cos(alpha)cos(beta)+sin(alpha)sin(beta)}}}
and
{{{cos(alpha+beta)= cos(alpha)cos(beta)-sin(alpha)sin(beta)}}}

to rewrite the denominator 

{{{
( (sin(2x)cos(x)+cos(2x)sin(x))+(sin(2x)cos(x)-cos(2x)sin(x)) )/
((cos(2x)cos(x)+sin(2x)sin(x))+(cos(2x)cos(x)-sin(2x)sin(x)) )}}}

Remove the parentheses

{{{
( sin(2x)cos(x)+cos(2x)sin(x)+sin(2x)cos(x)-cos(2x)sin(x) )/
(cos(2x)cos(x)+sin(2x)sin(x)+cos(2x)cos(x)-sin(2x)sin(x) )}}}

Cancel:

{{{
( sin(2x)cos(x)+cross(cos(2x)sin(x))+sin(2x)cos(x)-cross(cos(2x)sin(x)) )/
(cos(2x)cos(x)+cross(sin(2x)sin(x))+cos(2x)cos(x)-cross(sin(2x)sin(x)) )}}}

{{{
( sin(2x)cos(x)+sin(2x)cos(x) )/
(cos(2x)cos(x)+cos(2x)cos(x) )}}}


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

{{{
(2sin(2x)cos(x) )/
(2cos(2x)cos(x) )}}}

Cancel the 2'a and the cosines:

{{{
(cross(2)sin(2x)cross(cos(x)) )/
(cross(2)cos(2x)cross(cos(x)) )}}}

{{{sin(2x)/cos(2x)}}}

Use identity {{{sin(phi)/cos(phi)=tan(phi)}}}

{{{tan(2x)}}}

Edwin</pre>