Question

prove thate sinA+sin3A+sin5A\cosA+cos3A+cos5A=tan3A

Answer 1

I hope it helps u ...if any doubts u can ask me

Answer 2

Answer:

To prove : $\dfrac{\sin A+\sin3A+\sin5A}{\cos A+\cos3A+\cos5A}=\tan3A$

Consider LHS :

$\dfrac{\sin A+\sin3A+\sin5A}{\cos A+\cos3A+\cos5A}\\\\\\=\dfrac{\sin A+\sin 5A+\sin 3A}{\cos A+\cos5A+\cos3A}\\\\\\=\dfrac{2\sin\dfrac{A+5A}{2}\cos\dfrac{5A-A}{2}+\sin 3A}{2\cos\dfrac{A+5A}{2}\cos\dfrac{5A-A}{2}+\cos 3A}\\\\\\=\dfrac{2\sin3A\cos2A+\sin3A}{2\cos3A\cos2A+\cos3A}$

$=\dfrac{\sin3A(\cos2A+1)}{\cos3A(\cos2A+1)}\\\\\\=\dfrac{\sin3A}{\cos3A}\\\\\\=\tan3A$=R.HS.

Hence Proved.