Prove that: \[ \frac{\cos 3A + 2\cos 5A + \cos 7A} {\cos A + 2\cos 3A + \cos 5A} = \frac{\cos 5A}{\cos 3A} \]
Solution
L.H.S.
\[ = \frac{\cos 3A + 2\cos 5A + \cos 7A} {\cos A + 2\cos 3A + \cos 5A} \]Group first and third terms and use identity:
\[ \cos C + \cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2} \]
\[
=
\frac{
2\cos5A\cos2A + 2\cos5A
}{
2\cos3A\cos2A + 2\cos3A
}
\]
Take common factor:
\[ = \frac{ 2\cos5A(\cos2A+1) }{ 2\cos3A(\cos2A+1) } \]Cancel common factors:
\[ = \frac{\cos5A}{\cos3A} \]Hence Proved.