Question 1162367
{{{ e^(i4x) = cos(4x)+isin(4x) }}}
{{{ e^(i3x) = cos(3x)+isin(3x) }}} <br>

Solve for cos():
{{{  cos(4x) = e^(i4x)-isin(4x) }}}
{{{  cos(3x) = e^(i3x)-isin(3x) }}} <br>

Now (ignoring sin(4x)sin(3x) for the time being):

{{{ cos(4x)cos(3x) = (e^(i4x)-isin(4x))*(e^(i3x)-isin(3x)) }}}<br>


Which can be "simplified" to:
{{{ cos(4x)cos(3x) = e^(i4x)e^(i3x)-e^(i4x)isin(3x)-e^(i3x)isin(4x)-sin(4x)sin(3x) }}} <br>

and further expanded to:
{{{  cos(4x)cos(3x) = e^(i7x) - cos(4x)isin(3x) - i^2sin(4x)sin(3x) - cos(3x)isin(4x)-i^2sin(4x)sin(3x) - sin(4x)sin(3x) }}} <br>

{{{  cos(4x)cos(3x) = cos(7x)+isin(7x) -  cos(4x)isin(3x) - i^2sin(4x)sin(3x) - cos(3x)isin(4x)-i^2sin(4x)sin(3x) - sin(4x)sin(3x) }}}

{{{ cos(4x)cos(3x) = cos(7x)+isin(7x) -  cos(4x)isin(3x) - cos(3x)isin(4x) + sin(4x)sin(3x) }}}<br>

{{{  cos(4x)cos(3x) - sin(4x)sin(3x) = cos(7x)+isin(7x) -  cos(4x)isin(3x) - cos(3x)isin(4x) }}}<br>

{{{  cos(4x)cos(3x) - sin(4x)sin(3x) = cos(7x) + i(expression) }}}<br>

The part called "expression" must be zero since the left hand side is real, therefore:
{{{  cos(4x)cos(3x) - sin(4x)sin(3x) = cos(7x) }}}<br>


EDIT:  Fixed cos(4x) cut & paste error, it now reads properly