1. verify the following identity

please can someone help me on this problem.

$\displaystyle sinx+sin2x+sin3x+sin4x+sin5x divided by cosx+cos2x+cos3x+cos4x+cos5x$ = tan 3x

2. Hello,
Originally Posted by jab1
please can someone help me on this problem.

$\displaystyle sinx+sin2x+sin3x+sin4x+sin5x divided by cosx+cos2x+cos3x+cos4x+cos5x$ = tan 3x
$\displaystyle \sin(a)+\sin(b)=2 \sin \left(\frac{a+b}{2}\right)\cos \left(\frac{a-b}{2}\right)$

$\displaystyle \cos(a)+\cos(b)=2 \cos \left(\frac{a+b}{2}\right)\cos \left(\frac{a-b}{2}\right)$

$\displaystyle \sin(x)+\sin(2x)+\sin(3x)+\sin(4x)+\sin(5x)$
$\displaystyle =(\sin(x)+\sin(5x))+(\sin(2x)+\sin(4x))+\sin(3x)$
Using the formula :
$\displaystyle =2 \sin(3x) \cos(x)+2 \sin(3x) \cos(3x)+\sin(3x)$
$\displaystyle =\sin(3x) \left(2 \cos(x)+2 \cos(3x)+1\right)$

Similarly, we get :
$\displaystyle \cos(x)+\cos(2x)+\cos(3x)+\cos(4x)+\cos(5x)$
$\displaystyle =\cos(3x) \left(2 \cos(x)+2 \cos(3x)+1\right)$

Which simplifies

3. thank you very much