Take L.H.S.

cos A + cos 3A + cos 5A + cos 7A

= (cos 3A + cos A) + (cos 7A + cos 5A)

= 2 cos 2A cos A + 2 cos 6A cos A

= 2 cos A (cos 6A + cos 2A)

= 4 cos A cos 2A cos 4A

= R.H.S.

Hence Proved