# Math Help - verify the following identity

1. ## verify the following identity

please can someone help me on this problem.

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

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

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

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

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

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

Which simplifies

3. thank you very much